Choose the concept from the list above that best represents the item in each box. angle measure. segment measure

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1 Name lass ate 4-1 LL Support ongruent igures oncept List algebraic equation angle measure congruency statement congruent angles congruent polygons congruent segments congruent triangles proof segment measure hoose the concept from the list above that best represents the item in each box. 1. > ST 2. m/ congruency statement angle measure R Q S T congruent polygons 4. YZ 5 MN 5. n > nxyz 6. iven: is the angle bisector of /, and X M N congruent triangles is the perpendicular Y W bisector of. R P Prove: n > n Z congruent segments proof 7. m/ 5 5x m/w 5 x 1 28 Solve 5x 5 x 1 28 to find the measures of / and /W cm 9. / and /ST are vertical angles. So, segment measure / > /ST. congruent angles algebraic equation opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 1

2 Name lass ate 4-1 Think bout a Plan ongruent igures lgebra ind the values of the variables. M 3x Know 1. What do you know about the measure of each of the non-right angles? The measure of each of the non-right angles are complementary in. L 2t in. K KLM 2. What do you know about the length of each of the legs? ll of the legs are equal in length. 3. What types of triangles are shown in the figure? isosceles right triangles Need 4. What information do you need to know to find the value of x? You need to know that the measure of each of the non-right angles is What information do you need to know to find the value of t? You need to know that the length of each of the legs is 4 in. Plan 6. ow can you find the value of x? What is its value? nswers may vary. Sample: Set 3x equal to 45. So, 3x 5 45; x ow do you find the value of t? What is its value? nswers may vary. Sample: Set 2t equal to 4. So, 2t 5 4; t 5 2. opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 2

3 Name lass ate 4-1 Practice orm ongruent igures ach pair of polygons is congruent. ind the measures of the numbered angles M N S T L I R 135 P W 4 U K J Q V Y X ml ; ml6 5 90; ml ; ml ml3 5 90; ml ml7 5 40; ml kt OkJS. List each of the following. S 4. three pairs of congruent sides O JS, T O S, T O J 5. three pairs of congruent angles l OlJ, l OlS, lt Ol WXYZ O JKLM. List each of the following. 6. four pairs of congruent sides WZ O JM, WX O JK, XY O KL, ZY O ML 7. four pairs of congruent angles lw OlJ, lx OlK, ly OlL, lz OlM or xercises 8 and 9, can you conclude that the triangles are congruent? Justify your answers. W Z Y T X M L J J K 8. nj and nij Yes; lj OlIJ 9. nqrs and ntvs by Third ngles Thm. and by the Refl. Prop. R J O J. Therefore, J kj OkIJ by the ef. of O triangles. 95 S T Q 95 I V No; lqsr OlTSV because vert. angles are congruent, and lqrs OlTVS by Third ngles Thm., but none of the sides are necessarily congruent. 10. eveloping Proof Use the information given in the diagram. ive a reason that each statement is true. a. /L > /Q iven b. /LNM > /QNP Vert. angles are O. c. /M > /P Third ngles Thm. d. LM > QP, LN > QN, MN > PN iven e. nlnm > nqnp ef. of O triangles L P N M Q Prentice all old eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 3

4 Name lass ate 4-1 Practice (continued) orm ongruent igures or xercises 11 and 12, can you conclude that the figures are congruent? Justify your answers. 11. and No; answers may 12. n and njk vary. Sample: l does not have to K be a right angle. lgebra ind the values of the variables x 10 J Yes; answers may vary. Sample: l OlJ and l O K by the lt. Int. ngles Thm. and l OlJK by the Vert. ngles Thm., so all corresp. parts are congruent. 74 (5x) (3x 2) lgebra O J. ind the measures of the given angles or lengths of the given sides. 15. m/ 5 3y, m/ 5 y x 1 3; J 5 3x m/ 5 5z 1 20, m/ 5 6z b 1 4; J 5 3b LMNP > QRST. ind the value of x. 35 L M Q (5x) x P 45 T N R S (3x) 20. iven: is the angle bisector of /. is the perpendicular bisector of. Prove: n > n ecause is the angle bisector of l, l Ol. ecause is the perpendicular bisector of, O and l Ol. O by the Reflexive Property of ongruence. So, because the corresponding parts are all congruent, k Ok. Prentice all old eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 4

5 Name lass ate 4-1 Practice orm K ongruent igures ach pair of polygons is congruent. ind the measures of the numbered angles. 1. M Q L N P R 90; 40 95; I Use the diagram at the right for xercises 3 7. k OkXYZ. omplete the congruence statements. 3. > uxy To start, use the congruence statement to identify the points that correspond to and. Y X Z corresponds to u. X corresponds to u. Y 4. ZY > u 5. /Z > ul 6. / > z lyxz z 7. / > uly OUR O MNY. List each of the following. 8. four pairs of congruent angles l OlM; lo Ol; lu OlN; lr OlY 9. four pairs of congruent sides O O M; OU O N; UR O NY ; R O YM or xercises 10 and 11, can you conclude that the figures are congruent? Justify your answers. 10. nsrt and nprq 11. n and n S T R Q P No; lsrt OlPRQ because vert. ' are O, and lrst OlRPQ by Third ngles Thm., but none of the sides are necessarily congruent. Yes; l Ol by Third ngles Thm. Therefore, k Ok by the def. of O triangles. Prentice all oundations eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 5

6 Name lass ate 4-1 Practice (continued) orm K ongruent igures 12. iven: and bisect each other. > ; / > / Prove: n > n Statements Reasons 1) and bisect each other. 1) iven >, / > / 2) >, > 2) 9 efinition of bisect 3) / > / 3) 9 Vertical angles are O. 4) / > / 4) 9 Third ngles Theorem 5) n > n 5) 9 efinition of O Triangles 13. If n > njkl, which of the following must be a correct congruence statement? / > /L / > /K K J > JL n > nlkj L 14. Reasoning student says she can use the information in the figure to prove n > n. Is she correct? xplain. No; explanations may vary. Sample: orresponding parts are not congruent. The figure can be used to prove k Ok. lgebra ind the values of the variables. 15. nxyz > n 3; n > n 15 Z 30 (2y) (5x) 75 X 9 in. Y 3x in. lgebra k OkQRS. ind the measures of the given angles or the lengths of the given sides. 17. m/ 5 x 1 24; m/q 5 3x 36; x 2 2; RS 5 x ; 10 Prentice all oundations eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 6

7 Name lass ate 4-1 Standardized Test Prep ongruent igures Multiple hoice or xercises 1 6, choose the correct letter. 1. The pair of polygons at the right is congruent. What is m/j? J 2. The triangles at the right are congruent. Which of the following statements must be true? I / > / > / > / > 3. iven the diagram at the right, which of the following must be true? nxs > nxt nsx > nxt nxs > nxt nxs > nxt S X 4. If nrst > nxyz, which of the following need not be true? /R > /X /T > /Z RT > XZ SR > YZ I T 5. If n > n, m/ 5 50, and m/ 5 30, what is m/? If > QRST, m/ 5 x 2 10, and m/q 5 2x 2 30, what is m/? [2] l Ol and l Ol, both by the lt. Int ngles Thm. So, by Third ngles Thm., l Ol. ecause O by the Refl. Prop. of ongruence, and we know O and O, then all the corresponding parts are congruent and k Ok. [1] incomplete proof [0] no proof or incorrect proof Short Response 7. iven: 6, 6, >, > Prove: n > n opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 7

8 Name lass ate 4-1 nrichment ongruent igures Shared Implications Sometimes different statements share one or more implications. or example, QR ' ST and QR is the perpendicular bisector of ST share the implication that QR meets ST at a right angle. The statements below refer to the diagram at the right. 1. J ' JK ; 2. J ' ; 3. 6 JK ; 4. / > /K; 5. X > JX ; 6. > KJ ; X 7. K bisects J ; 8. J bisects K ; 9. m/ 5 m/j 5 90 J K Identify shared implications and reduce the number of given statements. 1. What implication is shared by Statement 5 and Statement 7? nswers may vary. Sample: oth statements imply that X O JX. 2. What implication is shared by Statement 3 and Statement 4? nswers may vary. Sample: oth statements imply that l OlK. 3. Which two statements share at least one implication with Statement 9? nswers may vary. Sample: Statements 1 and 2 4. an you prove nx > nkjx using only five of the statements above? If so, identify them, then complete the proof. Yes; Sample: statements 4, 6, 7, 8, and 9; K bisects J, so X O JX. J bisects K, so X O KX. O KJ is given, so all corresponding sides are congruent. lx is congruent to lkxj by the Vert. ngles Thm. ml 5 mlj 5 90 and l OlK are given, so all corresponding angles are congruent. So, kx OkKJX. 5. an you prove nx > nkjx using only four of the statements above? If so, identify them, then complete the proof. Yes; Sample: statements 6, 7, 8, and 3. K bisects J, so X O JX. J bisects K, so X O KX. O JK is given, so all corresp. sides are congruent. lx OlKXJ by the Vert. ngles Thm. ecause n JK, l OlJ, and l OlK, by the lt. Int. ngles Thm., all corresp. angles are congruent. So, kx OkKJX. 6. an you prove nx > nkjx using only three of the statements above if the only way to prove triangles congruent is through the definition of congruent triangles? If so, identify them, then complete the proof. no opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 8

9 Name lass ate 4-1 Reteaching ongruent igures iven > QRST, find corresponding parts using the names. Order matters. or example, QRST or example, QRST This shows that / corresponds to /Q. Therefore, / > /Q. This shows that corresponds to RS. Therefore, > RS. xercises ind corresponding parts using the order of the letters in the names. 1. Identify the remaining three pairs of corresponding angles and sides between and QRST using the circle technique shown above. l OlR, l OlS, l OlT, O QR, O ST, and O TQ ngles: Sides: QRST QRST QRST QRST QRST QRST 2. Which pair of corresponding sides is hardest to identify using this technique? nswers may vary. Sample: and QT ind corresponding parts by redrawing figures. 3. The two congruent figures below at the left have been redrawn at the right. Why are the corresponding parts easier to identify in the drawing at the right? K K nswers may vary. Sample: The drawing at the right shows figures in same orientation. 4. Redraw the congruent polygons at the right in the P same orientation. Identify all pairs of corresponding sides and angles. heck students work. l and T Q lp, l and lq, l and lr, l and ls, l and lt, and PQ, and QR, and RS, and ST, and and TP all correspond. S R 5. MNOP > QRST. Identify all pairs of congruent M N R S sides and angles. P O lm OlQ, ln OlR, lo OlS, lp OlT, MN O QR, NO O RS, OP O ST, and PM O TQ T Q opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 9

10 Name lass ate 4-1 Reteaching (continued) ongruent igures Problem iven n > n, m/ 5 30, and m/ 5 65, what is m/? ow might you solve this problem? Sketch both triangles, and put all the information on both diagrams m/ 5 30; therefore, m/ ow do you know? ecause / and / are corresponding parts of congruent triangles. 30 xercises Work through the exercises below to solve the problem above. 6. What angle in n has the same measure as /? What is the measure of that angle? dd the information to your sketch of n. l; You know the measures of two angles in n. ow can you find the measure of the third angle? nswers may vary. Sample: Use Triangle ngle-sum Thm. Set sum of all three angles equal to What is m/? ow did you find your answer? 85; answers may vary. Sample: ml ( ) efore writing a proof, add the information implied by each given statement to your sketch. Then use your sketch to help you with xercises dd the information implied by each given statement. 9. iven: / and / are right angles. ml 5 ml 5 90, ' and ' 10. iven: > and >. is a parallelogram because it has opposite sides that are congruent. 11. iven: / > /. n 12. an you conclude that / > / using the given information above? If so, how? Yes; use the Third ngles Thm. 13. ow can you conclude that the third side of both triangles is congruent? The third side is shared by both triangles and is congruent by the Refl. Prop. of ongruence. opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 10

11 Name lass ate 4-2 LL Support Triangle ongruence by SSS and SS Problem Use the figure at the right. ow can you prove that n > nxyz? Justify each step. iven: The figure at the right Prove: n > nxyz 1) > XY 1) iven 2) > YZ 2) iven 3) / > /X 3) iven 4) / > /Z 4) iven 5) / > /Y 5) Third ngles Theorem Z Y X 6) n > nxyz 6) Side-ngle-Side (SS) Postulate xercises 1. Use the figure at the right. ow can you prove that km OkTMS? Justify each step. iven: M is the midpoint of S and T. Prove: nm > ntms 1) M is the midpoint of S and T. 1) 2) M > TM 2) 3) M > SM 3) 4) /M > /TMS 4) 5) nm > ntms 5) iven efinition of the midpoint efinition of the midpoint Vertical angles are congruent. M Side-ngle-Side (SS) Postulate S T 2. Use the figure at the right. ow can you prove that ki OkJI? Justify each step. iven: is the midpoint of J. Prove: ni > nji 1) is the midpoint of J. 1) 2) > J 2) 3) I > I 3) 4) I > JI 4) 5) ni > nji 5) iven efinition of the midpoint Reflexive Property of O Third Side Postulate Side-Side Side (SSS) Postulate opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 11 I J

12 Name lass ate 4-2 Think bout a Plan Triangle ongruence by SSS and SS Use the istance ormula to determine whether n and n are congruent. Justify your answer. (1, 4), (5, 5), (2, 2) (25, 1), (21, 0), (24, 3) Understanding the Problem 1. You need to determine if n > n. What are the three ways you know to prove triangles congruent? If all corresponding parts are congruent, if all three sides are congruent, or, if two sides and the included angle are congruent, then the triangles are congruent. 2. What information is given in the problem? the coordinates for each vertex of each triangle Planning the Solution 3. If you use the SSS Postulate to determine whether the triangles are congruent, what information do you need to find? the lengths of the three sides of each triangle 4. ow can you find distances on a coordinate plane without measuring? Use the istance ormula. 5. In an ordered pair, which number is the x-coordinate? Which is the y-coordinate? The x-coordinate is the first number and the y-coordinate is the second number. etting an nswer 6. ind the length of each segment using the istance ormula, 5 "(y 1 2 y 2 ) 2 1 (x 1 2 x 2 ) 2. Your answers may be in simplest radical form. z "17 z z 3"2 z z "5 z z "17 z z 3"2 z z "5 z 7. Using the SSS Postulate, are the triangles congruent? xplain. Yes; the triangles are congruent, because three pairs of sides are congruent. opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 12

13 Name lass ate 4-2 Practice orm Triangle ongruence by SSS and SS raw kmt. Use the triangle to answer the questions below. 1. What angle is included between M and MT? lm 2. Which sides include /T? T and TM 3. What angle is included between T and M? l M T Would you use SSS or SS to prove the triangles congruent? If there is not enough information to prove the triangles congruent by SSS or SS, write not enough information. xplain your answer. 4. R K L N J Not enough information; two pairs of corresponding sides are congruent, but the congruent angle is not included. P R SS; two pairs of corresponding sides and their included angle are congruent. M O SSS; three pairs of corresponding sides are congruent. 7. Z 8. P L 9. X W Y Not enough information; two pairs of corresponding sides are congruent, but the congruent angle is not the included angle. N SSS; three corresponding sides are congruent. K SS; two pairs of corresponding sides and their included right angle are congruent. 10. O N 11. S 12. P R Not enough information; one pair of corresponding sides and corresponding angles are congruent, but the other pair of corresponding sides that form the included angle must also be congruent. T SS; two pairs of corresponding sides and their included vertical angles are congruent. Prentice all old eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 13 T SSS or SS; three pairs of corresponding sides are congruent, or, two pairs of corresponding sides and their included vertical angles are congruent. R

14 Name lass ate 4-2 Practice (continued) orm Triangle ongruence by SSS and SS 13. raw a iagram student draws n and nqrs. The following sides and angles are congruent: > QS > QR / > /R ased on this, can the student use either SSS or SS to prove that n > nqrs? If the answer is no, explain what additional information the student needs. Use a sketch to help explain your answer. Q No; l and lr are not the included angles for the sides given. To prove congruence, you would need to know either that O RS or lq Ol. R S 14. iven: >, > Prove: n > n Statements 1) O 2) O 3) l Ol 4) k Ok Reasons 1) iven 2) iven 3) Vertical ' are O. 4) SS 15. iven: WX 6 YZ, WX > YZ Prove: nwxz > nyzx Statements Reasons 1) WX n YZ 1) iven 2) lwxz OlYZX 2) lternate Interior ' are O. 3) WX O YZ 3) iven 4) ZX O XZ 4) Reflexive Property 5) kwxz OkYZX 5) SS W Z X Y 16. rror nalysis n and npqr are both equilateral triangles. Your friend says this means they are congruent by the SSS Postulate. Is your friend correct? xplain. Incorrect; both triangles being equilateral means that the three angles and sides of each triangle are congruent, but there is no information comparing the side lengths of the two triangles. 17. student is gluing same-sized toothpicks together to make triangles. She plans to use these triangles to make a model of a bridge. Will all the triangles be congruent? xplain your answer. Yes; because all the triangles are made from the same-sized toothpick, all three corresponding sides will be congruent. Prentice all old eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 14

15 Name lass ate 4-2 Practice orm K Triangle ongruence by SSS and SS 1. eveloping Proof opy and complete the flow proof. iven: RX > SX, QX > TX Prove: nqxr > ntxs Q R X S T RX SX and QX TX QXR TXS iven QXR SS TXS Vert. are. What other information, if any, do you need to prove the two triangles congruent by SS? xplain. To start, list the pairs of congruent, corresponding parts you already know Y S > u and > u Need l Ol; these are the included '. X Z R T u XZ > u RT and u ZY > u TS and u ly > uls need lz OlT or XY O RS Would you use SSS or SS to prove these triangles congruent? If there is not enough information to prove the triangles congruent by SSS or SS, write not enough information. xplain your answer SSS; third side is shared by both > and is O to itself by Refl. Prop. of O. SS; vertical ' are O. Prentice all oundations eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 15

16 Name lass ate 4-2 Practice (continued) orm K Triangle ongruence by SSS and SS Use the istance ormula to determine whether k and kjkl are congruent. Justify your answer. 6. (0, 0), (0, 4), (3, 0) To start, find the lengths of the corresponding sides. J(1, 4), K(23, 4), L(1, 1) 5 "(0 2 4) 2 1 (0 2 0) 2 5 u4 JK 5 "(4 2 4) 2 1 (12(23)) 2 5 u4 Yes; they are O by SSS. 5 u 5 KL 5 u 5 5 u 3 LJ 5 u3 7. (22, 5), (4, 23), (4, 3) J(2, 1), K(26, 7), L(26, 1) an you prove the triangles congruent? If so, write the congruence statement and name the postulate you would use. If not, write not enough information and tell what other information you would need. 8. R X 9. Q S W Yes; kqrs OkYXW by SS. No; they are not O because and JL have different lengths. 10. Reasoning Suppose >, / > /, and >. Is n congruent to n? xplain. Not necessarily; may not be O to. Y J K not enough information; need O K to apply SSS or l OlJ to apply SS 11. iven: is the perpendicular bisector of. Prove: n > n Statements Reasons 1) is the perpendicular bisector of. 1) iven 2) > 2) efinition of segment bisector 3) / and / are right '. 3) efinition of perpendicular 4) 9 l Ol 4) 9 ll right angles are O. 5) 9 O 5) 9 Reflexive Property of O 6) 9 k Ok 6) 9 SS Prentice all oundations eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 16

17 Name lass ate 4-2 Standardized Test Prep Triangle ongruence by SSS and SS Multiple hoice or xercises 1 4, choose the correct letter. 1. Which pair of triangles can be proved congruent by SSS? 2. Which pair of triangles can be proved congruent by SS? 3. What additional information do you need to prove nnop > nqsr? PN > SQ /P > /S NO > QR /O > /S P N O S Q R 4. What additional information do you need to prove ni > n? I > / > / I > I > I Short Response 5. Write a two-column proof. L iven: M is the midpoint of LS, PM > QM. Prove: nlmp > nsmq P [2] Statements: 1) M is the midpoint of LS; 2) LM O SM; 3) llmp OlSMQ; 4) PM O QM; 5) klmp OkSMQ; Reasons: 1) iven; 2) ef. of a midpoint; 3) Vert. ' are O; 4) iven; 5) SS [1] incomplete proof [0] incorrect or no proof M Q S opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 17

18 Name lass ate 4-2 nrichment Triangle ongruence by SSS and SS omplete each proof using the given information and figure J. iven: Rectangle J is a square. M 6 L 6 K 6 J 6 L J 6 I 6 J ' K L ' > L J > I > LR > T L K Z N P Q R S T Y W U X V 1. Prove: nk > nl Statements 1) L > KL > > 2) L 1 KL 5 K, 1 5 3) K > 4) L ', J ' K, L 6 K 5) L ' J, ' K 6) LK is a rectangle 7) L > K 8) m/k 5 90, m/l ) 9 lk OlL 10) nk > nl 2. Prove: njr > nt Statements 1) LR > T, > L 2) LR 1 R 5 L, T 1 T 5 3) 9 LR 1 R 5 T 1 T 4) 9 R O T 5) 6 K, 6 L, J 6 K 6) /T > /K, /K > /W 7) /W > /RJ 8) 9 lt OlRJ 9) J is a square. 10) 9 O J 11) njr > nt J I Reasons 1) 9 iven 2) 9 Segment ddition Postulate 3) 9 Substitution Property 4) 9 iven 5) 9 Perpendicular Transversal Theorem 6) efinition of a rectangle 7) Opposite sides of a rectangle are >. 8) 9 efinition of perpendicular lines 9) efinition of congruence 10) 9 SS Reasons 1) iven 2) 9 Segment ddition Postulate 3) Substitution Property 4) Segment Subtraction Postulate 5) 9 iven 6) 9 lternate interior angles are O. 7) 9 lternate interior angles are O. 8) Transitive Property of ongruence 9) 9 iven 10) efinition of a square 11) 9 SS opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 18

19 Name lass ate 4-2 Reteaching Triangle ongruence by SSS and SS You can prove that triangles are congruent using the two postulates below. Postulate 4-1: Side-Side-Side (SSS) Postulate If all three sides of a triangle are congruent to all three sides of another triangle, then those two triangles are congruent. J K Z If JK > XY, KL > YZ, and JL > XZ, then njkl > nxyz. In a triangle, the angle formed by any two sides is called the included angle for those sides. L Y X Postulate 4-2: Side-ngle-Side (SS) Postulate P Q If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle, then those two triangles are congruent. R If PQ >, PR >, and /P > /, then npqr > n. /P is included by QP and PR. / is included by and. xercises 1. What other information do you need to prove ntr > nr by SS? xplain. O TR; by the Reflexive Property of ongruence, R O R. It is given that ltr OlR. These are the included angles for the corresponding congruent sides. 2. What other information do you need to prove n > n by SS? xplain. l Ol; These are the included angles between the corresponding congruent sides. R T 3. eveloping Proof opy and complete the flow proof. iven: > M, J > Z Prove: nj > nzm J M M iven J Z iven J ZM SS Z J ZM Vertical are. opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 19

20 Name lass ate 4-2 Reteaching (continued) Triangle ongruence by SSS and SS Would you use SSS or SS to prove the triangles congruent? If there is not enough information to prove the triangles congruent by SSS or SS, write not enough information. xplain your answer. 4. P 5. J Y 6. X Q M Z M Y Not enough information; two pairs of corresponding Not enough Not enough information; only sides are congruent, but the information; you two pairs of corresponding congruent angles are not the need to know if sides are congruent. You need included angles. O Y. to know if O XY or lz Ol. 7. iven: PO > SO, O is the 8. iven: I >, ' I midpoint of NT. Prove: ni > n Prove: nnop > ntos P N O T Statements: 1) PO O SO; 2) O is the midpoint of NT ; 3) NO O TO ; 4) lnop OlTOS; 5) knop OkTOS; Reasons: 1) iven; 2) iven; 3) ef. of midpoint; 4) Vert. ' are O; 5) SS S 9. carpenter is building a support for a bird feeder. e wants the triangles on either side of the vertical post to be congruent. e measures and finds that > and that >. What would he need to measure to prove that the triangles are congruent using SS? What would he need to measure to prove that they are congruent using SSS? Statements: 1) O ; 2) I O, ' I; 3) l and li are rt. '; 4) l OlI; 5) ki Ok; Reasons: 1) Refl. Prop.; 2) iven; 3) ef. of perpendicular; 4) ll rt. ' are O; 5) SS I or SS, he would need to determine if l Ol; for SSS, he would need to determine if O. 10. n artist is drawing two triangles. She draws each so that two sides are 4 in. and 5 in. long and an angle is 558. re her triangles congruent? xplain. nswers may vary. Sample: Maybe; if both the 558 angles are between the 4-in. and 5-in. sides, then the triangles are congruent by SS. opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 20

21 Name lass ate 4-3 LL Support Triangle ongruence by S and S Problem Y iven: The figure at the right Prove: nq > nxyq Q X xplain Work Justify irst, list the information that is given directly in the diagram. Second, use the fact that all right angles are congruent to each other. Q > YQ / and /X are right '. / > /X iven ll right angles are congruent. Next, use the fact that vertical angles are congruent. /Q > /XQY Vertical ngles Theorem inally, determine which theorem can be used to prove the triangles congruent using the information listed above. nq > nxyq ngle-ngle-side (S) Theorem xercise iven: The figure at the right J Prove: njl > nklj K L xplain Work Justify irst, list the information that is given directly in the diagram. Second, use the fact that alternate interior angles are congruent when parallel lines are cut by a transversal. Next, recall that any line segment is congruent to itself. inally, determine which theorem can be used to prove the triangles congruent using the information listed above. J 6 KL L 6 KJ /JL > /KLJ /LJ > /KJL JL > LJ njl > nklj iven lternate Interior ngles Theorem Reflexive Property of ongruence ngle-side-ngle (S) Postulate opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 21

22 Name lass ate 4-3 Think bout a Plan Triangle ongruence by S and S iven: 6, 6 Prove: n > n 1. What do you need to find to solve the problem? Sample: at least three corresponding pairs of sides or angles that I can prove to be congruent 2. What are the corresponding parts of the two triangles? l and l; l and l; l and l; and ; and ; and and 3. What word would you use to describe? transversal 4. What can you show about angles in the triangles that can indicate congruency? I can find congruent angles using alternate interior angles of the transversal. 5. What do you know about a side or sides of the triangles that can be used to show congruency? The transversal is part of both triangles, so it is congruent to itself by the Reflexive Property of ongruence. 6. Write a proof in paragraph form. nswers may vary. Sample: n and n are given. Therefore is a transversal. l Ol and l Ol by the lternate Interior ngles Theorem. The transversal is part of both triangles, so O by the Reflexive Property of ongruence. k Ok by the S Postulate. opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 22

23 Name lass ate 4-3 Practice orm Triangle ongruence by S and S Name two triangles that are congruent by S. 1. I K M N 2. R X Y J L kij OkMLK O P U S T V krst OkYXZ Z W 3. eveloping Proof omplete the proof by filling in the blanks. K iven: /IJ > /KIJ /IJ > /IJK Prove: nij > nkij Proof: /IJ > /KIJ and /IJ > /IJK are given. IJ > IJ by 9. Refl. Prop. of ongruence So, nij > nkij by 9. S I J 4. iven: /LOM > /NPM, LM > NM L N Prove: nlom > nnpm Proof: llom OlNPM and LM O NM are given. llmo OlNMP because vert. ' are O. So, klom OkNPM by S. O M P 5. iven: / and / are right angles. bisects Prove: n > n Statements 1) l and l are right angles. 2) l Ol 3) l Ol 4) bisects 5) O 6) k Ok Reasons 1) iven 2) ll right angles are congruent. 3) Vertical angles are congruent. 4) iven 5) ef. of bisector 6) S Prentice all old eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 23

24 Name lass ate 4-3 Practice (continued) orm Triangle ongruence by S and S 6. eveloping Proof omplete the proof. K iven: /1 > /2, ' L, KL ' L, > KL Prove: n > nkl 1 2 L Proof: L is a right. 1 2 a. iven c. ' lines form right ' e. iven f. all right ' are O L f. KL L L is a right. KL b. iven d. ' lines form right ' g. iven KL h. S 7. Write a flow proof. 8. Write a two-column proof. J K iven: / > / iven: /K > /M / > / KL > ML L Prove: n > n Prove: njkl > npml M P iven Statements Reasons lk OlM iven KL O ML iven iven S Theorem ljlk OlPLM Vert. ' are O. kjkl OkPML S Postulate Reflexive Prop. of or xercises 9 and 10, write a paragraph proof. 9. iven: / > / 10. iven: JM bisects /J. > Prove: n > n l Ol is given. l Ol because vert. ' are O. O is given. So, k Ok by S. JM ' KL Prove: njmk > njml JM bisects lj is given. lkjm OlLJM by def. of an l bisector. JM O JM by the Refl. Prop. of O. JM ' KL is given. llmj and lkmj are right ' by the def. of perpendicular. Therefore, llmj OlKMJ because all right ' are O. So, kjmk OkJML by S. J K M L Prentice all old eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 24

25 Name lass ate 4-3 Practice orm K Triangle ongruence by S and S Name the two triangles that are congruent by S. 1. S V T L X Q U R Z Y J K krqs OkTUV k OkJKL R O N I M Q P ki OkMNP OkOQR 4. eveloping Proof omplete the two-column proof by filling in the blanks. iven: ', bisects / Prove: n > n Statements 1) ', bisects /. 2) 9 l and l are right '. 3) / > / 4) / > / 5) 9 O 6) 9 k Ok Reasons 1) iven 2) efinition of perpendicular 3) 9 ll right angles are O. 4) 9 efinition of l bisector 5) Reflexive Property of > 6) S 5. iven: KJ > MN, /KJL > /MNL Prove: njkl > nnml K L M Statements 1) KJ > MN, /KJL > /MNL 2) /KLJ > /MLN 3) 9 4) 9 llkj OlLMN kjkl OkNML Reasons J 1) iven 2) 9 Vertical ' are O. 3) Third ngles Theorem 4) S N Prentice all oundations eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 25

26 Name lass ate 4-3 Practice (continued) orm K Triangle ongruence by S and S 6. iven: PT > RS, /PTR > /RSP Prove: npqt > nrqs P Q R Statements Reasons T S 1) 9 PT O RS, lptr OlRSP 1) iven 2) /PQT > /RQS 3) 9 kpqt OkRQS 2) 9 3) S Vertical ' are O. 7. iven: is the angle bisector of / and /. Prove: n > n Statements 1) 9 2) 9 is the angle bisector of l and l. l Ol, l Ol 3) / > / 4) > 5) 9 k Ok Reasons 1) 9 iven 2) efinition of / bisector 3) 9 Third ngles Theorem 4) 9 Reflexive Property of O 5) S 8. Reasoning student tells you that he can prove the S Theorem using the SS Postulate and the Third ngles Theorem. o you agree with him? xplain. (int: ow many pairs of sides does the SS Postulate use?) No; answers may vary. Sample: the SS Postulate requires two pairs of corresp. O sides. 9. Reasoning an you prove the triangles congruent? Justify your answer. No; answers may vary. Sample. there are no included sides or included ' that correspond in both >, and you cannot use the Third ngles Theorem. Prentice all oundations eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 26

27 Name lass ate 4-3 Standardized Test Prep Triangle ongruence by S and S Multiple hoice or xercises 1 4, choose the correct letter. 1. Which pair of triangles can be proven congruent by the S Postulate? L R Q T U M N P S V W K X Y Z L J 2. or the S Postulate to apply, which side of the triangle must be known? the included side the shortest side the longest side a non-included side 3. Which pair of triangles can be proven congruent by the S Theorem? K J L O Q R M K P 4. or the S Theorem to apply, which side of the triangle must be known? the included side the longest side Short Response 5. Write a paragraph proof. iven: /3 > /5, /2 > /4 Prove: nvwx > nvyx W V Y the shortest side a non-included side [2] l3 Ol5 is given. l1 Ol5 because vert. ' are O. l3 Ol1 by the Trans. Prop. of O. l2 Ol4 X is given. VX O VX by the Refl. Prop. of O. kvwx OkVYX by S. [1] incomplete proof [0] incorrect or no proof opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 27 T S I

28 Name lass ate 4-3 nrichment Triangle ongruence by S and S ollow these steps to create a stunt plane from a sheet of in.-by-11-in. paper. 1. old the paper in half vertically. Using the definition of segment bisectors, note the congruent segments. heck students work. 2. old the upper corners to lie on either side of the center line. Using the definitions of angle bisectors and right angles, note the angles that are congruent. a. What theorem allows you to conclude that the two triangles you created are congruent? S b. Identify the congruent angles and sides. heck students work. 3. old the top point down along a line that is 1 in. below the bottom side of the triangles. gain, fold the upper corners to lie on either side of the center line. a. What theorem allows you to conclude that the two triangles you created are congruent? S b. Identify the congruent angles and sides. heck students work. 4. old the small triangle up along the line formed by the bottom sides of the triangles formed in Step 3. This exposes two more congruent triangles. The two triangles are congruent by S. a. xplain which angles are congruent and why. heck students work. b. Name the congruent sides, and explain how you know the sides are congruent. The included sides are congruent because the fold in Step 1 found the midpoint of the width of the paper, thus creating two equal segments. 5. old along the vertical fold line formed in Step 1 so that the Step 5 triangles formed in Step 4 are on the outside of the airplane. 6. Turn the paper so that the long edge is at the bottom. To complete the paper airplane, fold each top edge down to meet the bottom edge, and crease. Step 2 Step 3 ollow the Steps to create a dart plane from a sheet of in.-by-11-in. paper. 7. Repeat Steps 1 and 2 above. 8. old the upper side corners to lie on either side of the center line. Step 8 a. What theorem allows you to conclude that the two triangles you created are congruent? S b. Identify the congruent angles and sides. heck students work. 9. Turn the paper over. old the outer edges in so that they lie on either side of the center line. 10. old the paper in half along the original fold made in Step djust the wing flaps until they are at right angles with the body to complete the dart. opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 28

29 Name lass ate 4-3 Reteaching Triangle ongruence by S and S Problem an the S Postulate or the S Theorem be applied directly to prove the triangles congruent? R a. ecause /R and / are right angles, they are congruent. > by the Reflexive Property of >, and it is given that /R > /. Therefore, nr > n by the S Theorem. xercises Indicate congruences. 1. opy the top figure at the right. Mark the figure with the angle congruence and side congruence symbols that you would need to prove the triangles congruent by the S Postulate. 2. opy the second figure shown. Mark the figure with the angle congruence and side congruence symbols that you would need to prove the triangles congruent by the S Theorem. b. It is given that > and / > /. ecause / and / are vertical angles, they are congruent. Therefore, n > n by the S Postulate. 3. raw and mark two triangles that are congruent by either the S Postulate or the S Theorem. heck students work. What additional information would you need to prove each pair of triangles congruent by the stated postulate or theorem? S Postulate 5. S Theorem J K 6. S Postulate l Ol ljmk OlLKM, lzxy OlZVU ljkm OlLMK, ljmk OlLMK, or M ljkm OlLKM L 7. S Theorem 8. S Theorem P 9. S Postulate ly OlO Y O lp Ol R T lyl OlLY X U Y Y Z V L opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 29

30 Name lass ate 4-3 Reteaching (continued) Triangle ongruence by S and S 10. Provide the reason for each step in the two-column proof. iven: TX 6 VW, TU > VU, /XTU > /WVU, /UWV is a right angle. Prove: ntux > nvuw Statements 1) /UWV is a right angle. 2) VW ' XW 3) TX 6 VW 4) TX ' XW 5) /UXT is a right angle. 6) /UWV > /UXT 7) TU > VU 8) /XTU > /WVU 9) ntux > nvuw Reasons 1) 9 2) 9 5) 9 6) 9 7) 9 8) 9 iven 9) 9 S Theorem T X efinition of perpendicular lines 3) 9 iven 4) 9 Perpendicular Transversal Theorem efinition of perpendicular lines ll right angles are congruent. iven iven U V W 11. Write a paragraph proof. iven: WX 6 ZY ; WZ 6 XY Prove: nwxy > nyzw W Z It is given that WX n ZY and WZ n XY, so lxwy OlZYW and lxyw OlZWY, by the lternate Interior ' Thm. X Y WY O YW by the Reflexive Property of O. So, by S Post. kwxy OkYZW. 12. eveloping Proof omplete the proof by filling in the blanks. iven: / > /, /1 > /2 1 Prove: n > n Proof: / > / and /1 > /2 are given. > by So, n > n by 9. S Refl. Prop. of ongruence 3 P 13. Write a paragraph proof. 1 6 iven: /1 > /6, /3 > /4, LP > OP 2 5 Prove: nlmp > nonp L M 4 N O 3 l3 Ol4 is given. Therefore, ml3 5 ml4, by def. of Ols. ecause l2 and l3 are linear pairs, and l4 and l5 are linear pairs, the pairs of angles are suppl. Therefore, l2 Ol5 by the ongruent Suppl. Thm. l1 Ol6 and LP O OP are given, so klmp OkONP, by the S Thm. opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 30

31 Name lass ate 4-4 LL Support Using orresponding Parts of ongruent Triangles There are two sets of note cards below that show how to prove is the perpendicular bisector of. The set on the left has the statements and the set on the right has the reasons. Write the statements and the reasons in the correct order. Statements Reasons l Ol efinition of the perpendicular bisector O ngle-ngle-side (S) Theorem k Ok is the perpendicular bisector of. O ; l and l are right angles; 6 l Ol When parallel lines are cut by a transversal, alternate interior angles are congruent. orresponding parts of congruent triangles are congruent. iven ll right angles are congruent. Statements Reasons 1) O ; l and l are right angles; n 2) l Ol 1) 2) iven ll right angles are congruent. 3) l Ol 3) When parallel lines are cut by a transversal, alternate interior angles are congruent. 4) k Ok 4) ngle-ngle-side (S) Theorem 5) O 5) orresponding parts of congruent triangles are congruent. 6) is the perpendicular bisector of. 6) efinition of the perpendicular bisector opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 31

32 Name lass ate 4-4 Think bout a Plan Using orresponding Parts of ongruent Triangles onstructions The construction of / congruent to given / is shown. > because they are the radii of the same circle. > because both arcs have the same compass settings. xplain why you can conclude that / > /. Understanding the Problem 1. What is the problem asking you to prove? l Ol 2. Segments and are not drawn on the construction. raw them in. What figures are formed by drawing these segments? two triangles 3. What information do you need to be able to use corresponding parts of congruent triangles? needs to be shown as congruent to. Planning the Solution 4. To use corresponding parts of congruent triangles, which two triangles do you need to show to be congruent? k and k 5. What reason can you use to state that >? They are congruent because they are radii of the same circle by construction. etting an nswer 6. Write a paragraph proof that uses corresponding parts of congruent triangles to prove that / > /. O and O because they are both radii of the same circle. O because they both have the same compass settings. Therefore, k Ok by SSS and l Ol by PT. opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 32

33 Name lass ate 4-4 Practice orm Using orresponding Parts of ongruent Triangles or each pair of triangles, tell why the two triangles are congruent. ive the congruence statement. Then list all the other corresponding parts of the triangles that are congruent. 1. M lmkl OlKJ 2. N P J because vertical angles are congruent, so kkj OkKLM by K S. lkml OlKJ, R Q L MK O K, and LK O JK. 3. omplete the proof. Y Z PR O RP because the shared side of the two triangles is congruent to itself, so kprq OkRPN by SSS. lprn OlRPQ, lnpr OlQRP, and lrnp OlPQR. iven: Y >, / > /Y Prove: Z > Statements Reasons 1) Y >, / > /Y 1) 9 iven 2) /YZ and / are vertical angles. 2) efinition of vertical angles 3) /YZ > / 3) 9 4) 9 kzy Ok 4) 9 5) 9 Z O 5) 9 Vertical angles are congruent. S PT 4. Open-nded onstruct a figure that involves two congruent triangles. Set up given statements and write a proof that corresponding parts of the triangles are congruent. heck students work. Prentice all old eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 33

34 Name lass ate 4-4 Practice (continued) orm Using orresponding Parts of ongruent Triangles 5. omplete the proof. iven: ', ', > Prove: / > / Statements Reasons 1) ', ' 1) 9 iven 2) / and / are right angles. 2) efinition of right angles 3) / > / 3) 9 ll right angles are congruent. 4) 9 l Ol 4) Vertical angles are congruent. 5) > 5) 9 iven 6) 9 k Ok 6) 9 S 7) / > / 7) 9 PT 6. onstruction Use a construction to prove that the two base angles of an isosceles triangle are congruent. iven: Isosceles n with base Prove: / > / Statements Reasons 1) n is isosceles. 1) 9 iven 2) > 2) efinition of isosceles triangle. 3) onstruct the midpoint of and call it. onstruct. 3) onstruction 4) 9 O 4) efinition of midpoint 5) > 5) 9 Refl. Prop. of ongruence 6) n > n 6) 9 SSS 7) 9 l Ol 7) 9 PT Prentice all old eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 34

35 Name lass ate 4-4 Practice orm K Using orresponding Parts of ongruent Triangles 1. eveloping Proof State why the two triangles are congruent. Then list all other corresponding parts of the triangles that are congruent. S; kqrs OkTWX; lq OlT, RS O WX Q T R S W X 2. eveloping Proof State why nxy and nyx are congruent. Then list all other corresponding parts of the triangles that are congruent. nswers may vary. Sample: SS; lyx OlXY, Y O X 3. iven: QS 6 RT, /R > /S Prove: /QTS > /TQR X Y Q R To start, determine how you can prove the triangles are congruent. The triangles share a side and have a pair of congruent angles. ecause QS 6 u, RT alternate interior angles /SQT and z lrtq are congruent. The triangles can be proven congruent by S. z S T Statements 1) 9 QS 6 RT, lr OlS 2) 9 lsqt OlRTQ 3) 9 QT O TQ 4) 9 kstq OkRQT 5) 9 lqts OlTQR Reasons 1) iven 2) lternate interior ' are >. 3) Reflexive Property of ongruence 4) S 5) orresp. parts of > > are >. Reasoning opy and mark the figure to show the given information. xplain how you would prove O. 4. iven: >, / > / S and orresp. parts of O> are O. 5. iven: bisects, bisects SS and orresp. parts of O> are O. 6. iven: 6, 5 nswers may vary. Sample: S and orresp. parts of O> are O. Prentice all oundations eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 35

36 Name lass ate 4-4 Practice (continued) orm K Using orresponding Parts of ongruent Triangles 7. iven: K is the perpendicular bisector of. Prove: > Statements 1) K is the perpendicular bisector of. 2) 9 K O K 3) /K > /K 4) 9 K O K 5) nk > nk 6) 9 O Reasons 1) 9 iven 2) ef. of perpendicular bis. 3) ef. of perpendicular bis.; all right ' are >. 4) Refl. Prop. of > 5) 9 SS K 6) orresp. parts of > > are >. 8. iven: is a rectangle. is the midpoint of. Prove: > Statements 1) is a rectangle. is the midpoint of. 2) / > / 3) > 4) 9 5) 9 6) 9 O k Ok O Reasons 1) iven 2) efinition of rectangle 3) efinition of rectangle 4) 9 efinition of midpoint 5) 9 SS 6) 9 orresp. parts of O> are O. Prentice all oundations eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 36

37 Name lass ate 4-4 Standardized Test Prep Using orresponding Parts of ongruent Triangles Multiple hoice or xercises 1 6, choose the correct letter. 1. ased on the given information in the figure at the right, how can you justify that nj > nji? S S SSS S J I 2. In the figure at the right the following is true: / > / and / > /. ow can you justify that n > n? SS S SSS PT 3. nrm > nkyz. ow can you justify that YZ > RM? PT SS S SSS 4. Which statement cannot be justified given only that npj > ntim? P > TI I / > /I /JP > /IMT JP > MI 5. In the figure at the right, which theorem or postulate can you use to prove nm > nzm? S SS SSS S 6. In the figure at the right, which theorem or postulate can you use to prove nk > n? S SS SSS S Z M K Short Response 7. What would a brief plan for the following proof look like? iven: >, / > / Prove: > [2] O by the Reflexive Property. k Ok by SS. O by PT; [1] one step missing or one reason incorrect [0] incorrect or no response opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 37

38 Name lass ate 4-4 nrichment Using orresponding Parts of ongruent Triangles String rt rtists have always used geometry, but in some cases geometry can become the driving force in art. In string art, for example, polygons combine to make interesting patterns. To make string art, you usually start with a frame with nails at equal intervals. The frame below left was used to make the art on the right. (The string wrapping around the nails has been omitted.) Use the diagram and information below for xercise 1. iven: PQ 5 QR 5 RS 5 ST 5 TU 5 UV 5 VW 5 WX 5 XY 5 YZ U ramework for attaching strings Q R S T V I W J X Y P Z 1. ind as many triangles as you can prove congruent in the above diagram. Name the postulate or theorem (SSS, SS, S, or S) that justifies your answer. nswers may vary. Sample: ktuz OkVUP (SS); ksuy OkWUQ (SS); kqp OkYJZ (S); ktjs OkVW (SSS) Use the diagram and information below for xercises 2 and 3. iven: Q 5 R, Q ' SV ' R, 6 2. ind three pairs of congruent triangles in the diagram. or each pair of triangles, name the postulate or theorem (SSS, SS, S, or S) that justifies your answer. Sample: kqw OkWR (S); kq OkR (SS); kw OkWQ (S) 3. If 6 6 and 6 6, find as many congruent triangles as you can in the diagram, and give an example for each type of triangle. Sample: 7 > congruent to kijn, 23 > congruent to kw, 15 > congruent to kq, and 7 > congruent to ks M S I N Q J T K W U L O R Y V P opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 38

39 Name lass ate 4-4 Reteaching Using orresponding Parts of ongruent Triangles If you can show that two triangles are congruent, then you can show that all the corresponding angles and sides of the triangles are congruent. Problem iven: 6, / > / Prove: > In this case you know that 6. forms a transversal and creates a pair of alternate interior angles, / and /. You have two pairs of congruent angles, / > / and / > /. ecause you know that the shared side is congruent to itself, you can use S to show that the triangles are congruent. Then use the fact that corresponding parts are congruent to show that >. ere is the proof: Statements Reasons 1) 6 1) iven 2) / > / 2) lternate Interior ngles Theorem 3) / > / 3) iven 4) > 4) Reflexive Property of ongruence 5) n > n 5) S 6) > 6) PT xercises 1. Write a two-column proof. M iven: MN > MP, NO > PO Prove: /N > /P P N O Statements Reasons 1) 9 MN O MP, NO O PO 1) iven 2) MO > MO 2) 9 3) 9 kmno OkMPO 3) 9 Reflexive Property of O SSS 4) /N > /P 4) 9 PT opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 39

40 Name lass ate 4-4 Reteaching (continued) Using orresponding Parts of ongruent Triangles 2. Write a two-column proof. iven: PT is a median and an altitude of nprs. Prove: PT bisects /RPS. Statements 1) PT is a median of nprs. 2) 9 T is the midpoint of RS. 3) 9 RT O ST 4) PT is an altitude of nprs. 5) PT ' RS 6) /PTS and /PTR are right angles. 7) 9 lpts OlPTR 8) 9 PT O PT 9) 9 kpts OkPTR 10) /TPS > /TPR 11) 9 PT bisects lrps. Reasons 1) 9 iven 2) efinition of median 3) efinition of midpoint 4) 9 iven 5) 9 efinition of altitude 6) 9 efinition of perpendicular 7) ll right angles are congruent. 8) Reflexive Property of ongruence 9) SS 10) 9 PT 11) 9 efinition of angle bisector 3. Write a two-column proof. iven: QK > Q; Q bisects /KQ. Prove: K > K Q Statements Reasons 1) QK O Q; Q bisects lkq. 1) iven 2) lkq OlQ 2) ef. of l bis. 3) Q O Q 3) Refl. Prop. of ongruence 4) kkq OkQ 4) SS 5) K O 5) PT 4. Write a two-column proof. O iven: ON bisects /JO, /J > / Prove: JN > N J N Statements Reasons 1) ON bisects ljo, lj Ol 1) iven 2) ljon OlON 2) ef. of l bis. 3) ON O ON 3) Refl. Prop. of ongruence 4) kjon OkON 4) S 5) JN O N 5) PT opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 40

41 Name lass ate 4-5 LL Support Isosceles and quilateral Triangles omplete the vocabulary chart by filling in the missing information. Word or Word Phrase base efinition The base of an isosceles triangle is the side included between the pair of congruent angles. Picture or xample base angles 1. The base angles are the two congruent angles of an isosceles triangle. corollary 2. corollary is a theorem that can be proved easily by another theorem. corollary to the Isosceles Triangle Theorem is: If a triangle is equilateral, then the triangle is equiangular. equiangular triangle isosceles triangle n equiangular triangle is a triangle with three congruent angles. The angles of an equiangular triangle all measure n isosceles triangle is a triangle with two congruent sides. 3. legs The legs are the two congruent sides of an isosceles triangle. 5. vertex angle 6. The vertex angle is the 7. angle formed by the legs of an isosceles triangle. opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 41

42 Name lass ate 4-5 Think bout a Plan Isosceles and quilateral Triangles lgebra The length of the base of an isosceles triangle is x. The length of a leg is 2x 2 5. The perimeter of the triangle is 20. ind x. Know 1. What is the perimeter of a triangle? the sum of the sides of a triangle 2. What is an isosceles triangle? a triangle with two sides the same length Need 3. What are the sides of an isosceles triangle called? base and leg 4. ow many of each type of side are there? 1 base and 2 legs 5. The lengths of the base and one leg are given. What is the third side of the triangle called? leg Plan 6. Write an expression for the length of the third side. 2x Write an equation for the perimeter of this isosceles triangle. x 1 (2x 2 5) 1 (2x 2 5) Solve the equation for x. Show your work. 5x x 5 30 x 5 6 opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 42

43 Name lass ate 4-5 Practice orm Isosceles and quilateral Triangles omplete each statement. xplain why it is true. 1. / > 9 > / l; all the angles of an equilateral triangle are congruent. 2. / > 9 l; the base angles of an isosceles triangle are congruent. 3. / > 9 > / l; all the angles of an equilateral triangle are congruent. 4. > 9 > ; all the sides of an equilateral triangle are congruent. lgebra ind the values of x and y ; ; y y (y 10) 55; x x 135 (x 5) 8. 30; ; x 110 y 45; 45 3y 2x y x Use the properties of isosceles and equilateral triangles to find the measure of the indicated angle. 11. m/ m/ m/ quilateral n and isosceles n share side. If m/ 5 34 and 5, what is the measure of /? (int: it may help to draw the figure described.) 172 Prentice all old eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 43

44 Name lass ate 4-5 Practice (continued) orm Isosceles and quilateral Triangles Use the diagram for xercises to complete each congruence statement. xplain why it is true. 15. > 9 ; onverse of the Isosceles Triangle Theorem 16. > 9 ; onverse of the Isosceles Triangle Theorem 17. > 9 ; onverse of the Isosceles Triangle Theorem 18. The wall at the front entrance to the Rock and Roll all of ame and Museum in leveland, Ohio, is an isosceles triangle. The triangle has a vertex angle of 102. What is the measure of the base angles? Reasoning n exterior angle of an isosceles triangle has the measure 130. ind two possible sets of measures for the angles of the triangle. 50, 50, and 80; 50, 65, and Open-nded raw a design that uses three equilateral triangles and two isosceles triangles. Label the vertices. List all the congruent sides and angles. heck students work. lgebra ind the values of m and n ; n n 44; ; m n n m m Writing xplain how a corollary is related to a theorem. Use examples from this lesson in making your comparison. theorem is a statement that is proven true by a series of steps. corollary is a statement that can be taken directly from the conclusion of a theorem, usually by applying the theorem to a specific situation. or example, Theorems 4-3 and 4-4 are general statements about all isosceles triangles. Their corollaries apply the theorems to equilateral triangles. Prentice all old eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved

45 Name lass ate 4-5 Practice orm K Isosceles and quilateral Triangles omplete each statement. xplain why it is true. 1. > 9 nswers may vary. Sample: ; the legs of an isosceles triangle are congruent. 2. / > 9 nswers may vary. Sample: l; the base angles of an isosceles triangle are congruent. 3. / > 9 > / nswers may vary. Sample: l; all the angles of an equilateral triangle are congruent. lgebra ind the values of x and y ; 30 To start, determine what types of triangles are shown x in the diagram. Then use an equation to find x. ecause two sides are marked congruent in both 45 triangles, the triangles are both 9. isosceles y u 1 x 5 u 5. 15; y 69; 37 (4x) y x 37 x Use the properties of isosceles triangles to complete each statement. 7. If m/ 5 54, then m/ If 5 8, then You are asked to put a V-shaped roof on a house. The slope of the roof is 408. What is the measure of the angle needed at the vertex of the roof? Reasoning The measure of one angle of a triangle is 30. Of the two remaining angles, the larger angle is four times the size of the smaller angle. Is the triangle isosceles? xplain. Yes, because the measure of the smaller angle is 30. Prentice all oundations eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 45

46 Name lass ate 4-5 Practice (continued) orm K Isosceles and quilateral Triangles or xercises 11 and 12, use the diagram to complete each congruence statement. Then list the theorem or corollary that proves the statement. The first one has been done for you. / > 9 nswer: / (or /); orollary to Theorem > 9 nswers may vary. Sample: or ; orollary to Theorem / > 9 nswers may vary. Sample: l or l; orollary to Theorem 4-3 or xercises 13 15, use the diagram to complete each congruence statement. Then list the theorem or corollary that proves the statement. 13. PR > /RUV > 9 QR; onverse of the Isosceles Triangle Theorem lrvu; Isosceles Triangle Theorem P S U 15. SR > 9 TR; onverse of the Isosceles Triangle Theorem Q T V R 16. Reasoning n equilateral triangle and an isosceles triangle share a common side as shown at the right. What is the measure of the vertex angle? xplain. 120; the congruent angles in the diagram both have a measure of 60. The base angles of the isosceles triangle have a measure of 30 because one is the other angle in a right triangle. The vertex angle must measure 120 if the base angles both measure lgebra ind the values of m and n m n n 130; 105 m 67.5; 45 Prentice all oundations eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 46

47 opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 47 Name lass ate 4-5 Standardized Test Prep Isosceles and quilateral Triangles ridded Response Solve each exercise and enter your answer on the grid provided. Refer to the diagram for xercises What is the value of x? 2. What is the value of y? 3. What is the value of z? 4. The measures of two of the sides of an equilateral triangle are 3x 1 15 in. and 7x 2 5 in. What is the measure of the third side in inches? 5. In ni, I 5, m/i 5 3x 1 4, and m/i 5 2x What is m/i? 125 y x z nswers

48 Name lass ate 4-5 nrichment Isosceles and quilateral Triangles The swan below is composed of several triangles. Use the given information and the figure to find each angle measure. Note: igure not drawn to scale. iven: n is equilateral; / > /; > > ; / > /; n > nk > njm; nk > nkl; KO > O; /KN > /NK ; JN > JO (4x) (4y) (7x) (6z) (10z) (3z) (16y) (2z 6) (8x 2) (4z 7) I J (4y) K L M (x 6) N (8y) O 1. m/ m/ m/ m/ 5. m/ m/ m/ m/ 9. m/ m/ m/ m/ 13. m/ m/ m/ m/k 17. m/k m/k m/k m/k 21. m/k m/kl m/kl m/kl 25. m/jm m/mj m/jm m/ok m/ok m/ko m/kn m/nk m/okn m/jno m/jon m/njo 100 opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 48

49 Name lass ate 4-5 Reteaching Isosceles and quilateral Triangles Two special types of triangles are isosceles triangles and equilateral triangles. n isosceles triangle is a triangle with two congruent sides. The base angles of an isosceles triangle are also congruent. n altitude drawn from the shorter base splits an isosceles triangle into two congruent right triangles. n equilateral triangle is a triangle that has three congruent sides and three congruent angles. ach angle measures 608. You can use the special properties of isosceles and equilateral triangles to find or prove different information about a given figure. Look at the figure at the right. You should be able to see that one of the triangles is 40 equilateral and one is isosceles. y Problem What is m/? 2 x 2 n is isosceles because it has two base angles that are congruent. ecause the sum of the measures of the angles of a triangle is 180, and m/ 5 40, you can solve to find m/. Problem m/ 1 m/ 1 m/ There are 1808 in a triangle. m/ m/ Substitution Property 2 m/ ombine like terms. 2 m/ Subtraction Property of quality m/ 5 70 ivision Property of quality What is? n is equilateral because it has three congruent angles. 5 (2 1 2) 5 4, and 5 5. So, 5 4. opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 49

50 Name lass ate 4-5 Reteaching (continued) Isosceles and quilateral Triangles Problem What is the value of x? 40 ecause x is the measure of an angle in an equilateral triangle, x Problem y x 2 2 What is the value of y? m/ 1 m/ 1 m/ There are 1808 in a triangle y Substitution Property y 5 50 Subtraction Property of quality xercises omplete each statement. xplain why it is true. 1. / > 9 l; base angles of an isosceles triangle are congruent. 2. / > 9 > / l; the angles of an equilateral triangle are congruent. 3. > 9 > ; the sides of an equilateral triangle are congruent. etermine the measure of the indicated angle. 4. / / / 55 lgebra ind the value of x and y. 7. y x 35; ; 50 y x 9. Reasoning n exterior angle of an isosceles triangle has a measure 140. ind two possible sets of measures for the angles of the triangle. 40, 40, 100; 40, 70, 70 opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 50

51 Name lass ate 4-6 LL Support ongruence in Right Triangles The column on the left shows the steps used to prove that O. Use the column on the left to answer each question in the column on the right. Problem O in Right Triangles iven: is the midpoint of and. Prove: O 1) is the midpoint of and ; 6 ; m/ 5 90 iven 2) > ; > efinition of midpoint 3) / > / lternate Interior ngles Theorem 4) / and / are right angles. efinition of a right angle 5) n > n ypotenuse-leg (L) Postulate 1. What is the definition of the midpoint of a line segment? The midpoint is halfway between the two endpoints. It divides the line segment into two equal halves. 2. ow do you know that 6 and m/ 5 90? by reading the diagram 3. What does the symbol > between two line segments mean? The line segments are congruent. 4. What does the word interior mean? Interior means on the inside of. 5. What is the measure of /? What information is necessary to apply the L Postulate? Two right triangles must have congruent hypotenuses and one congruent leg. 6) > orresponding parts of congruent triangles are congruent. 7. What are the corresponding angles and sides for n and n? l and l, l and l, l and l, and, and, and opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 51

52 Name lass ate 4-6 Think bout a Plan ongruence in Right Triangles lgebra or what values of x and y are the triangles congruent by L? Know 3y x y x y 5 x 5 1. or two triangles to be congruent by the ypotenuse-leg Theorem, there must be a pair of right angles, and the lengths of the hypotenuses and one of the legs of each triangle must be equal. 2. The length of the hypotenuse of the triangle on the left is 3y 1 x and the hypotenuse of the triangle on the right is y The length of the leg of the triangle on the left is y 2 x and the length of the leg of the triangle on the right is x 1 5. Need 4. To solve the problem you need to find the values of x and y. Plan 5. What system of equations can you use to find the values of x and y? 3y 1 x 5 y 1 5; y 2 x 5 x What method(s) can you use to solve the system of equations? Sample: Solve one equation for y and substitute into the other equation; graph both equations. 7. What is the value of y? What is the value of x? 3; 21 opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 52

53 Name lass ate 4-6 Practice orm ongruence in Right Triangles 1. eveloping Proof omplete the paragraph proof. iven: RT ' SU, RU > RS. Prove: nrut > nrst Proof: It is given that RT ' SU. So, lrts and lrtu are right R angles because perpendicular lines form right angles. RT > RT by the Reflexive Property of ongruence. It is given that RU > RS. So, nrut > nrst by L. S T U 2. Look at xercise 1. If m/rst 5 46, what is m/rut? Write a flow proof. Use the information from the diagram to prove that n > n. and are right angles. iven iven nswers may vary. Sample: and are right. efinition of right L Theorem Reflexive Property of ongruence 4. Look at xercise 3. an you prove that n > n without using the ypotenuse-leg Theorem? xplain. Yes; answers may vary. Sample: You know that O from the diagram and O by the Reflexive Property of ongruence. ecause the triangles are right triangles, the sides are related by the Pythagorean Theorem. If we let the legs 5 x and the hypotenuses 5 y, then the length of the other leg will be "y 2 2 x 2 on both triangles. So, by SSS k Ok. onstruct a triangle congruent to each triangle by the ypotenuse-leg Theorem Prentice all old eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 53

54 Name lass ate 4-6 Practice (continued) orm ongruence in Right Triangles lgebra or what values of x or x and y are the triangles congruent by L? x 17 x ; x 1 2x 1 x 3 2; 1 2x y 8x 2y x y 3x 3y 11. Write a paragraph proof. iven: bisects, > ; /, / are right angles. Prove: n > n k and k are right triangles because each contains a right angle (definition of a right triangle). It is given that O, so the hypotenuses of these right triangles are congruent. ecause bisects, point is the midpoint of. O (definition of a midpoint), so the triangles have a pair of congruent legs. y L, k Ok. What additional information would prove each pair of triangles congruent by the ypotenuse-leg Theorem? 12. R Q 13. S l and l are right angles. l and lq are right angles. 14. L X 15. T TR O TV M LN O XZ N Y Z 16. Reasoning re the triangles congruent? xplain. No; they are both right triangles, and one pair of legs is congruent, but the hypotenuse of one triangle is congruent to a leg of the other triangle. R S V Prentice all old eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 54

55 Name lass ate 4-6 Practice orm K ongruence in Right Triangles 1. eveloping Proof omplete the proof. iven: /WVZ and /VWX are right angles. WZ > VX Prove: nwvz > nvwx To prove that right triangles nwvz and nvwx are congruent, you must prove that the hypotenuses are congruent and that one 9 is congruent. leg Statements 1) 9 2) 9 3) 9 4) 9 lwvz and lvwx are right '. WZ O VX WV O VW kwvz OkVWX Reasons 1) iven 2) iven 3) Reflexive Property of ongruence 4) L Theorem V Z Y W X 2. Look at xercise 1. If m/x 5 54, what is m/z? Look at xercise 1. If m/x 5 54, what is m/vwz? Study xercise 1. an you prove that nwvz and nvwx are congruent without using the L Theorem? xplain. Yes; answers may vary. Sample: by using the Pythagorean Theorem to find another pair of congruent sides lgebra or what values of x and y are the triangles congruent by L? 5. x 4 6. x 2y y 5 2y x 14; 7 y 3 y x x 7 7. Reasoning The LL Theorem says that two right triangles are congruent if both pairs of legs are congruent. What theorem or postulate could be used to prove that the LL Theorem is true? xplain. nswers may vary. Sample: The SS Postulate could be used to prove the LL Theorem because the two right triangles would have two pairs of congruent legs and the included right angles would also be congruent. 2; 7 Prentice all oundations eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 55

56 Name lass ate 4-6 Practice (continued) orm K ongruence in Right Triangles What additional information would prove each pair of triangles congruent by the ypotenuse-leg Theorem? T N S R L M O lm and ls are right angles. 10. J M 11. W X L O Z Y K N WZ O YX or WX O YZ JL O MO or KL O NO oordinate eometry Use the figure at the right for ( 3, 5) xercises omplete the paragraph proof that shows that and are perpendicular. 4 2 The slope of is u. 22 The slope of is u. 0.5 The product of the two slopes is u. 21 Therefore the line segments are 9. perpendicular y (1, 3) 4 (1, 3) x 13. ow do you know that 5? ( 3, 5) The istance ormula shows that both line segments have length 2" omplete the paragraph proof below that shows that n > n. right angle / is a 9 from xercise 12. You can also use the product of slopes to show that / is a right angle. n and n share the same hypotenuse. You can use the 9 to show that 5. Therefore, by the 9, n > n. istance ormula L Theorem Prentice all oundations eometry Teaching Resources opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 56

57 Name lass ate 4-6 Standardized Test Prep ongruence in Right Triangles Multiple hoice or xercises 1 4, choose the correct letter. 1. Which additional piece of information would allow you to prove that the triangles are congruent by the L theorem? m/ 5 40 m/ 5 m/ > > 2. or what values of x and y are the triangles shown congruent? x 5 1, y 5 4 x 5 4, y 5 1 x 5 2, y 5 4 x 5 1, y 5 3 2x 8 10x y 2x 10x y 3. Two triangles have two pairs of corresponding sides that are congruent. What else must be true for the triangles to be congruent by the L Theorem? The included angles must be right angles. They have one pair of congruent angles. oth triangles must be isosceles. There are right angles adjacent to just one pair of congruent sides. 4. Which of the following statements is true? n > ni by SS. n > ni by SS. n > n by L. n > ni by L. I xtended Response 5. re the given triangles congruent by the L Theorem? xplain. [4] No; they are right triangles, and have a pair of congruent legs ( O ), but the hypotenuses, and, are not congruent. So, the triangles only meet two of the three conditions for congruence by the L Theorem. [3] appropriate response plus a discussion of two of the three criteria for congruence [2] recognition only that the hypotenuses are not congruent [1] recognition that the triangles are not congruent [0] incorrect or no response opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 57

58 Name lass ate 4-6 nrichment ongruence in Right Triangles Right Triangle Patterns n art student wants to make a painting with a simple geometric pattern. She starts with a square. She divides this square into two congruent triangles. Then she divides each of these triangles into two smaller congruent triangles. She repeats the process seven more times. What does her pattern look like in the end? 1. Show that the two triangles are congruent using the ypotenuse-leg Theorem. Sample: ach is a right triangle. They have at least one pair of congruent legs and they have congruent hypotenuses. 2. Use your knowledge of the ypotenuse-leg Theorem to divide each triangle in the figure above into two smaller congruent triangles. Repeat the process six more times. heck students work. 3. ow do you know that the triangles at each step are congruent? Sample: ach is a right triangle, with equal legs and hypotenuses. 4. ow many triangles of the smallest size are shown? ow many triangles are shown if they each contain 64 of the smallest-sized unit? ow many triangles are shown if they each contain nine of the smallest-sized unit? hallenge ind the sizes of all 16 different-sized triangles in the diagram. 1, 2, 4, 8, 9, 16, 18, 25, 32, 36, 49, 50, 64, 72, 98, 128 opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 58

59 Name lass ate 4-6 Reteaching ongruence in Right Triangles Two right triangles are congruent if they have congruent hypotenuses and if they have one pair of congruent legs. This is the ypotenuse-leg (L) Theorem. P Q R n > npqr because they are both right triangles, their hypotenuses are congruent ( > PR), and one pair of legs is congruent ( > QR). Problem ow can you prove that two right triangles that have one pair of congruent legs and congruent hypotenuses are congruent (The ypotenuse-leg Theorem)? oth of the triangles are right triangles. / and / are right angles. > and >. ow can you prove that n > n? Look at n. raw a ray starting at that passes through. Mark a point X so that X 5. Then draw X to create nx. See that X >. (You drew this.) /X > /. (ll right angles are congruent.) >. (This was given.) So, by SS, n > nx. X X > (by PT) and >. (This was given.). So, by the Transitive Property of ongruence, X >. Then, /X > /. (ll right angles are congruent.) y the Isosceles Theorem, /X > /. So, by S, nx > n. Therefore, by the Transitive Property of ongruence, n > n. Problem re the given triangles congruent by the ypotenuse-leg Theorem? If so, write the triangle congruence statement. / and / are both right angles, so the triangles are both right. I > I by the Reflexive Property and I > is given. So, ni > ni. I opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 59

60 Name lass ate 4-6 Reteaching (continued) ongruence in Right Triangles xercises etermine if the given triangles are congruent by the ypotenuse-leg Theorem. If so, write the triangle congruence statement. 1. T 2. R L S T U V M N not congruent 3. S 4. L Z krsz OkTSZ O P Q M N klmn OkRVS V R X Y Measure the hypotenuse and length of the legs of the given triangles with a ruler to determine if the triangles are congruent. If so, write the triangle congruence statement Z kopq OkZYX J M k OkM k OkIJ I 7. xplain why nlmn > nomn. Use the ypotenuse-leg Theorem. ecause lnml and lnmo are right angles, both triangles are right triangles. It is given that their hypotenuses are congruent. ecause they share a leg, one pair of corresponding legs is congruent. ll criteria are met for the triangles to be congruent by the ypotenuse-leg Theorem. N L M O 8. Visualize n and n, where 5 and 5. What else must be true about these two triangles to prove that the triangles are congruent using the ypotenuse-leg Theorem? Write a congruence statement. l and l are right angles, or l and l are right angles. k Ok or k Ok. opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 60

61 Name lass ate 4-7 LL Support ongruence in Overlapping Triangles student wanted to prove O given l Ol and O. She wrote the statements and reasons on note cards, but they got mixed up. O because corresponding parts of congruent triangles are congruent. l Ol because vertical angles are congruent. O by the Reflexive Property of ongruence. k Ok by the ngle-ngle- Side (S) Theorem. l Ol and O are given. l Ol because corresponding parts of congruent triangles are congruent. k Ok by the Side-ngle- Side (SS) Theorem. Use the note cards to write the steps in order. 1. irst, l Ol and O are given. 2. Second, O by the Reflexive Property of ongruence. 3. Third, k Ok by the Side-ngle-Side (SS) Theorem. 4. Next, l Ol because corresponding parts of congruent triangles are congruent. 5. Then, l Ol because vertical angles are congruent. 6. Then, k Ok by the ngle-ngle-side (S) Theorem. 7. inally, O because corresponding parts of congruent triangles are congruent. opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 61