Name Class Date. This shows that A corresponds to Q. Therefore, A Q. This shows that BC corresponds to RS. Therefore, BC RS.

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1 ame lass ate Reteaching ongruent igures Given 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. jr, js, ST, 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? G H K G K H 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. j and jp, j and jq, j and jr, j and js, j and jt, T Q and PQ, and QR, and RS, and ST, and and TP all correspond. S R 5. OP QRST. Identify all pairs of congruent R S sides and angles. P O jq, jr, js, QR, RS, ST, and TQ T Q opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved.

2 ame lass ate Reteaching (continued) ongruent igures Problem Given, m = 30, and m = 65, what is m? How might you solve this problem? Sketch both triangles, and put all the information on both diagrams m = 30; therefore, m = 30. How 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 has the same measure as? What is the measure of that angle? dd the information to your sketch of. j; You know the measures of two angles in. How 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? How did you find your answer? 85; answers may vary. Sample: mj = 180 ( ) 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. Given: and are right angles. mj = mj = 90, # and # 10. Given: and. is a parallelogram because it has opposite sides that are congruent. 11. Given:. } 12. an you conclude that using the given information above? If so, how? Yes; use the Third ngles Thm. 13. How 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.

3 ame lass ate 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 JKL XYZ. 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 PQR. P is included by QP and PR. is included by and. xercises 1. What other information do you need to prove TR R by SS? TR; by the Reflexive Property of ongruence, R. It is given that jr. These are the included angles for the corresponding congruent sides. R T 2. What other information do you need to prove by SS? xplain. j; These are the included angles between the corresponding congruent sides. 3. eveloping Proof opy and complete the flow proof. Given:, J Z Prove: J Z Z Given J Z Given J Z SS Vertical are. jz opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved.

4 ame lass ate 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 G Z Y ot enough information; two pairs of corresponding ot enough ot 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. Y. to know XY or j. 7. Given: PO SO, O is the 8. Given: HI HG, H # GI midpoint of T. Prove: HI HG Prove: OP TOS P O T Statements: 1) SO; 2) O is the midpoint of T; 3) TO; 4) jtos; 5) TOS; Reasons: 1) Given; 2) Given; 3) ef. of midpoint; 4) Vert. ; 5) SS S 9. carpenter is building a support for a bird feeder. He wants the triangles on either side of the vertical post to be congruent. He 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? G H Statements: 1) H; 2) HG, H # GI; 3) jhg and jhi are rt. ; 4) jhi; 5) HG; Reasons: 1) Refl. Prop.; 2) Given; 3) ef. of perpendicular; 4) ll rt. ; 5) SS I or SS, he would need to determine if j; for SSS, he would need to determine 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 55. re her triangles congruent? xplain. nswers may vary. Sample: aybe; if both the 55 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.

5 ame lass ate Reteaching Triangle ongruence by S and S Problem an the S Postulate or the S Theorem be applied directly to prove the triangles congruent? H R a. ecause R and are right angles, they are congruent. by the Reflexive Property of, and it is given that R. Therefore, R by the S Theorem. xercises Indicate congruences. 1. opy the top figure at the right. ark 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. ark 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 H H and. ecause H and H are vertical angles, they are congruent. Therefore, H H 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 j jlk, jzvu jlk, jlk, or jlk L 7. S Theorem 8. S Theorem P 9. S Postulate jo Y O j R T jly G X U Y Y Z V L opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved.

6 ame lass ate Reteaching (continued) Triangle ongruence by S and S 10. Provide the reason for each step in the two-column proof. Given: TX } VW, TU VU, XTU WVU, UWV is a right angle. Prove: TUX VUW Statements 1) UWV is a right angle. 2) VW # XW 3) TX } VW 4) TX # XW 5) UXT is a right angle. 6) UWV UXT 7) TU VU 8) XTU WVU 9) TUX VUW Reasons 1)? Given 2)? efinition of perpendicular lines 3)? Given 4)? Perpendicular Transversal Theorem 5)? efinition of perpendicular lines 6)? ll right angles are congruent. 7)? 8)? 9)? Given Given T X S Theorem U V W 11. Write a paragraph proof. Given: WX } ZY ; WZ } XY Prove: WXY YZW W Z It is given that WX } ZY and WZ } XY, so jzyw and jzwy, by the lternate Interior Thm. X Y YW by the Reflexive Property So, by S Post. YZW. 12. eveloping Proof omplete the proof by filling in the blanks. Given:, Prove: Proof: and 1 2 are given. by?. 4 2 So, by?. S Refl. Prop. of ongruence 3 P 13. Write a paragraph proof. 1 6 Given: 1 6, 3 4, LP OP 2 5 Prove: LP OP L 4 O 3 j4 is given. Therefore, mj3 mj4, by def. js. ecause j2 and j3 are linear pairs, and j4 and j5 are linear pairs, the pairs of angles are suppl. Therefore, j5 by the ongruent Suppl. Thm. j6 and OP are given, so OP, by the S Thm. opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved.

7 ame lass ate 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 Given: }, Prove: In this case you know that }. 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. Here is the proof: Statements Reasons 1) } 1) Given 2) 2) lternate Interior ngles Theorem 3) 3) Given 4) 4) Reflexive Property of ongruence 5) 5) S 6) 6) PT xercises 1. Write a two-column proof. Given: P, O PO Prove: P P O Statements Reasons P, PO 1) Given 2) O O 2)? Reflexive Property 3)? PO 3)? SSS 4) P 4)? PT opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved.

8 ame lass ate Reteaching (continued) Using orresponding Parts of ongruent Triangles 2. Write a two-column proof. Given: PT is a median and an altitude of PRS. Prove: PT bisects RPS. Statements 1) PT is a median of PRS. 2)? T is the midpoint of RS. 3)? ST 4) PT is an altitude of PRS. 5) PT # RS 6) PTS and PTR are right angles. 7)? jptr 8)? PT 9)? PTR 10) TPS TPR 11)? PT bisects jrps. Reasons 1)? Given 2) efinition of median 3) efinition of midpoint 4)? Given 5)? efinition of altitude 6)? efinition of perpendicular 7) ll right angles are congruent. 8) Reflexive Property of ongruence 9) SS 10)? PT 11)? efinition of angle bisector 3. Write a two-column proof. Given: QK Q; Q bisects KQ. Prove: K K Q Statements Reasons 1) Q; Q bisects jkq. 1) Given 2) jq 2) ef. of j bis. 3) Q 3) Refl. Prop. of ongruence 4) Q 4) SS 5) 5) PT 4. Write a two-column proof. O Given: O bisects JOH, J H Prove: J H J H Statements Reasons 1) O bisects jjoh, jh 1) Given 2) jho 2) ef. of j bis. 3) O 3) Refl. Prop. of ongruence 4) HO 4) S 5) H 5) PT opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved.

9 ame lass ate 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. ote: igure not drawn to scale. Given: is equilateral; ; ; G G; G GK JH; KH KLH; KO O; HK HK ; J JO (4x) (4y) (7x) (6z) (10z) (3z) G (16y) (2z 6) H (8x 2) (4z 7) I J (4y) K L (x 6) (8y) O 1. m m m m m m m m m m m m m G m G m G m KG m KG m GK m KH m HK m KH m KHL m HKL m KLH m HJ m HJ m JH m OK m OK m KO m HK m HK m OK m JO m JO m JO 100 opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved.

10 ame lass ate 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 = 60. Problem y x 2 2 G What is the value of y? m + m + m = 180 There are 180 in a triangle y = 180 Substitution Property y = 50 Subtraction Property of quality xercises omplete each statement. xplain why it is true. 1.? j; base angles of an isosceles triangle are congruent. 2.? j; the angles of an equilateral triangle are congruent. 3. G? G; the sides of an equilateral triangle are congruent. etermine the measure of the indicated angle G G 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.

11 ame lass ate 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 Hypotenuse-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 Hypotenuse-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. How do you know that the triangles at each step are congruent? Sample: ach is a right triangle, with equal legs and hypotenuses. 4. How many triangles of the smallest size are shown? How many triangles are shown if they each contain 64 of the smallest-sized unit? How 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.

12 ame lass ate Reteaching (continued) ongruence in Right Triangles xercises etermine if the given triangles are congruent by the Hypotenuse-Leg Theorem. If so, write the triangle congruence statement. 1. T 2. L R S T U V not congruent Z TSZ 3. S 4. O L P Q RVS V R X Y Z ZYX easure 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 G H HIJ I 7. xplain why L O. Use the Hypotenuse-Leg Theorem. ecause jl and jo 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 Hypotenuse-Leg Theorem. L O 8. Visualize and, where = and =. What else must be true about these two triangles to prove that the triangles are congruent using the Hypotenuse-Leg Theorem? Write a congruence statement. j and j are right angles, or j and j are right opyright by Pearson ducation, Inc., or its affiliates. ll Rights Reserved.

Name Class Date. This shows that A corresponds to Q. Therefore, A Q. This shows that BC corresponds to RS. Therefore, BC RS.

Name Class Date. This shows that A corresponds to Q. Therefore, A Q. This shows that BC corresponds to RS. Therefore, BC RS. Name lass ate eteaching ongruent igures iven, find corresponding parts using the names. rder matters. or example, or example, his shows that corresponds to. herefore,. his shows that corresponds to. herefore,.