4-1. Congruent Figures. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary. 1. Underline the correct word to complete the sentence.

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1 4-1 ongruent igures Vocabulary Review 1. Underline the correct word to complete the sentence. polygon is a two-dimensional figure with two / three or more segments that meet exactly at their endpoints. 2. ross out the figure(s) that are NOT polygons. Vocabulary uilder congruent (adjective) kahng GROO unt Main Idea: ongruent figures have the same size and shape. Related Word: congruence (noun) Use Your Vocabulary 3. ircle the triangles that appear to be congruent. Write T for true or for false. 4. ongruent angles have different measures. 5. prism and its net are congruent figures. opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. T 6. The corresponding sides of congruent figures have the same measure. hapter 4 90

2 Key oncept ongruent igures ongruent polygons have congruent corresponding parts their matching sides and angles. When you name congruent polygons, you must list corresponding vertices in the same order. 7. Use the figures at the right to complete each congruence statement. G GH H > > G > GH > H / > l / > l / > lg / > lh Problem 1 Using ongruent Parts Got It? If kwys OkMKV, what are the congruent corresponding parts? 8. Use the diagram at the right. raw an arrow from each vertex of the first triangle to the corresponding vertex of the second triangle. WYS MKV 9. Use the diagram from xercise 8 to complete each congruence statement. Sides WY > MK YS > KV WS > MV ngles /W > lm /Y > lk /S > lv Problem 2 inding ongruent Parts opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. Got It? Suppose that kwys OkMKV. If m/w 5 62 and m/y 5 35, what is mlv? xplain. Use the congruent triangles at the right. 10. Use the given information to label the triangles. Remember to write corresponding vertices in order. 11. omplete each congruence statement. /W > lm /Y > /S > lk lv 12. Use the Triangle ngle-sum theorem. 35 W 62 m/s 1 m lw 1 m ly 5 180, so m/s ( ), or omplete. Since /S > lv and m/s 5 83, m/v Y S V K M 91 Lesson 4-1

3 Problem 3 inding ongruent Triangles Got It? Is k Ok? Justify your answer. 14. Underline the correct word to complete the sentence. To prove two triangles congruent, show that all adjacent / corresponding parts are congruent. 15. ircle the name(s) for n. acute isosceles right scalene 16. ross out the congruence statements that are NOT supported by the information in the figure. > > > / > / / > / / > / 17. You need 6 congruence statements to prove two triangles congruent, so you can / cannot prove that n > n. Theorem 4 1 Third ngles Theorem Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. Use k and k above. 18. If m/ 5 74, then m/ If m/ 5 44, then m/ If m/ 5 62, then m/ Problem 4 Proving Triangles ongruent Got It? Given: l Ol, O, O, O Prove: k Ok If... / > / and / > / 21. You are given four pairs of congruent parts. ircle the additional information you need to prove the triangles congruent. third pair second pair third pair of congruent of congruent of congruent sides angles angles Then... / > / opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. hapter 4 92

4 22. omplete the steps of the proof. 1) >, >, > 1) Given 2) / > l 2) Given 3) / > l 3) Vertical angles are congruent. 4) / > l 4) Third ngles Theorem 5) n > k 5) efinition of > triangles Lesson heck o you UNRSTN? If each angle in one triangle is congruent to its corresponding angle in another triangle, are the two triangles congruent? xplain. 23. Underline the correct word to complete the sentence. To disprove a conjecture, you need one / two / many counterexample(s). 24. n equilateral triangle has three congruent sides and three 608angles. ircle the equilateral triangles below. opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. 25. Use your answers to xercise 24 to answer the question. No. xplanations may vary. Sample: quilateral triangles have all pairs of corresponding angles congruent, but not all equilateral triangles are congruent. Math Success heck off the vocabulary words that you understand. congruent Need to review polygons Rate how well you can identify congruent polygons Now I get it! 93 Lesson 4-1

5 4-2 Triangle ongruence by SSS and SS Vocabulary Review 1. Use the diagram at the right. ind each. included angle between and l included side between / and / included angle between and included side between / and / included angle between and included side between / and / Vocabulary uilder l l postulate (noun) PHS chuh lit efinition: postulate is a statement that is accepted as true without being proven true. Main Idea: In geometry, you use what you know to be true to prove new things true. The statements that you accept as true without proof are called postulates or axioms. Use Your Vocabulary 2. Underline the correct word to complete the sentence. You can use properties, postulates, and previously proven theorems as reasons / statements in a proof. 3. Multiple hoice What is a postulate? a convincing argument using deductive reasoning a conjecture or statement that you can prove true a statement accepted as true without being proven true opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. a conclusion reached by using inductive reasoning hapter 4 94

6 Postulate 4 1 Side-Side-Side (SSS) Postulate Postulate 4-1 Side-Side-Side (SSS) Postulate If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent. 4. Use the figures at the right to complete the sentence. If >, >, and >, then n > n. Problem 1 Using SSS Got It? Given: O, O Prove: k Ok 5. You know two pairs of sides that are congruent. What else do you need to prove the triangles congruent by SSS? a third pair of congruent sides 6. The triangles share side. 7. omplete the steps of the proof. Statement Reason 1) > 1) Given 2) > 2) Given opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. 3) > 3) Reflexive Property of > 4) n > k 4) SSS Postulate 4 2 Side-ngle-Side (SS) Postulate Postulate 4 2 Side-ngle-Side (SS) Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. Use the figures below to complete each statement. 8. n > ktrs by SS. R S T 9. n > k by SSS. 95 Lesson 4-2

7 Problem 2 Using SS Got It? What other information do you need to prove kl OkNL by SS? L 10. ircle the angles that are marked congruent in the diagram. /L /L /NL /NL N 11. ircle the sides that form the angles that are marked congruent in the diagram. L N L L LN 12. omplete each congruence statements. L > L /L > lln Underline the correct word(s) to complete each sentence. 13. Proving nl > nnl by SS requires one / two pair(s) of congruent sides and one / two pair(s) of congruent angles. 14. The diagram shows congruency of zero / one / two pair(s) of congruent sides and zero / one / two pair(s) of congruent angles. 15. To prove the triangles congruent by SS, you still need zero / one / two pair(s) of congruent sides and zero / one / two pair(s) of congruent angles. 16. To prove the triangles congruent, you need to prove N and L congruent. Problem 3 Identifying ongruent Triangles Got It? Would you use SSS or SS to prove the triangles below congruent? xplain. omplete each statement with SSS or SS. 17. Use 9 if you have three pairs of sides congruent. 18. Use 9 if you have two pairs of sides and the included angle congruent. Write T for true or for false. T T 19. The diagram shows congruence of three sides. 20. In the triangle on the left, the marked angle is the included angle of the side with two marks and the side with three marks. 21. In the triangle on the right, the marked angle is the included angle of the side with two marks and the side with three marks. SSS SS opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. hapter 4 96

8 22. Would you use SSS or SS to prove the triangles congruent? xplain. SSS. xplanations may vary. Sample: You can t use SS because the included angles aren t between congruent sides. Lesson heck o you UNRSTN? rror nalysis Your friend thinks that the triangles below are congruent by SS. Is your friend correct? xplain. 23. re two pairs of corresponding sides congruent? Yes / No 24. Is there a pair of congruent angles? Yes / No 25. re the congruent angles the included angles between the corresponding congruent sides? 26. re the triangles congruent by SS? xplain. Yes / No opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. No. xplanations may vary. Sample: They cannot be proven congruent by SS because the congruent angles are not included angles. Math Success heck off the vocabulary words that you understand. congruent Need to review corresponding Rate how well you can use SSS and SS to prove triangles congruent Now I get it! 97 Lesson 4-2

9 4-3 Triangle ongruence by S and S Vocabulary Review 1. ross out the figure(s) that are NOT triangle(s). 2. triangle is a polygon with 3 sides. 3. triangle with a right angle is called a(n) obtuse / right / scalene triangle. Vocabulary uilder corresponding (adjective) kawr uh SPHN ding corresponding sides X Other Word orms: correspond (verb); correspondence (noun) efinition: orresponding means similar in position, purpose, or form. Math Usage: ongruent figures have congruent corresponding parts. Use Your Vocabulary raw a line from each part of k in olumn to the corresponding part of kxyz in olumn. olumn 4. /Z 5. / /Y 6. YZ 7. / /X olumn Y Z Z Y X opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. 8. XY 9. / XZ hapter 4 98

10 Postulate 4 3 ngle-side-ngle (S) Postulate Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. If... / > /, >, / > / Then... n > n 10. xplain how the S Postulate is different from the SS Postulate. xplanations may vary. Sample: The S Postulate uses two angles and an included side and the SS Postulate uses two sides and an included angle. Problem 1 Using S Got It? Which two triangles are congruent by S? xplain. H O N G I T opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. 11. Name the triangles. List the vertices in corresponding order: list the vertex with the one arc first, the vertex with the two arcs second, and the third vertex last. kgho 12. /G > / I > / 13. /H > / N > / 14. HG > NI > 15. The congruent sides that are included between congruent angles are HG and. 16. Write a congruence statement. Justify your reasoning. n GHO > n T kin kt nswers may vary. Sample: kgho and kt have two pairs of congruent angles and a pair of included congruent sides. 99 Lesson 4-3

11 Problem 2 Writing a Proof Using S Got It? Given: l Ol, O, l and l are right angles Prove: k Ok 17. omplete the flow chart to prove n > n. Given Given Given / and / are right angles. / / ll right angles are congruent. / / S Postulate n n Theorem 4 2 ngle-ngle-side (S) Theorem Theorem If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the two triangles are congruent. If... / > /, / > /, > Then... n > n 18. The nonincluded congruent sides of n and n are and. Problem 3 Writing a Proof Using S Got It? Given: ls OlQ, RP bisects lsrq Prove: ksrp OkQRP 19. How do you know which angles in the diagram are corresponding angles? nswers may vary. Sample: They have the same number of arcs. 20. omplete the statements to provensrp > nqrp. Statements Reasons 1) /S > lq 1) Given 2) RP bisects lsrq 2) Given 3) /SRP > lqrp 3) efinition of an angle bisector S P R Q opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. 4) RP > RP 4) Reflexive Property of ongruence 5) nsrp > lqrp 5) S hapter 4 100

12 Problem 4 etermining Whether Triangles re ongruent Got It? re kpr and ksir congruent? xplain. P I 21. The congruence marks show that / > li and PR > SR. 22. What other corresponding congruent parts exist? xplain. R S lpr OlSRI because they are vertical angles. 23. re npr and nsir congruent? If so, what theorem proves them congruent? Yes. S. Lesson heck o you UNRSTN? Reasoning Suppose l OlI and O GI. What else must you know in order to prove k and kghi are congruent by S? y S? 24. Label the diagram at the right. 25. To prove the triangles congruent by S, what do you need? two pairs of congruent angles and a I H G pair of included congruent sides 26. To prove the triangles congruent by S, what do you need? opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. two pairs of congruent angles and a pair of nonincluded congruent sides 27. If you want to use S, / and / G must also be congruent. 28. If you want to use S, / and / H must also be congruent. Math Success heck off the vocabulary words that you understand. included nonincluded corresponding Rate how well you can use S and S. Need to review Now I get it! 101 Lesson 4-3

13 4-4 Using orresponding Parts of ongruent Triangles Vocabulary Review Underline the correct word(s) to complete each sentence. 1. The Reflexive Property of ongruence states that any geometric figure is congruent / similar to itself. 2. The Reflexive Property of quality states that any quantity is equal to / greater than / less than itself. 3. ircle the expressions that illustrate the Reflexive Property of quality. a 5 a If 5 2, then (x 1 y) 5 3x 1 3y 5 1 c c 4. ircle the expressions that illustrate the Reflexive Property of ongruence. If / > /, then / > /. If > LM and LM > XY, then > XY. Vocabulary uilder proof (noun) proof / > / Related Word: prove (verb) > efinition: proof is convincing evidence that a statement or theory is true. Math Usage: proof is a convincing argument that uses deductive reasoning. Use Your Vocabulary omplete each statement with proof or prove. 5. In geometry, a 9 uses definitions, postulates, and theorems to prove theorems. 6. No one can 9 how our universe started. 7. He can 9 when he bought the computer because he has a receipt. proof prove prove opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. hapter 4 102

14 8. omplete the steps in the proof. Given: >, >, / > /, / > / Prove: n > n Statements Reasons 1) > > 1) Given 2) > 2) Reflexive Property of > 3) / > l / > l 3) Given 4) / > l 4) Third ngles Theorem 5) n > k 5) efinition of > triangles Problem 1 Proving Parts of Triangles ongruent Got It? Given: O, O Prove: l Ol 9. Name four ways you can use congruent parts of two triangles to prove that the triangles are congruent. SSS SS S S opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. 10. To prove triangles are congruent when you know two pairs of congruent corresponding sides, you can use SSS or SS. Underline the correct word to complete the sentence. 11. The Given states and the diagram shows that there are one / two / three pairs of congruent sides. 12. Give a reason for each statement of the proof. Statements 1) > 1) 2) > 2) 3) / > / 3) 4) n > n 4) 5) / > / 5) Reasons Given Given Vertical angles are congruent. SS orresponding parts of congruent triangles are congruent. 103 Lesson 4-4

15 Problem 2 Proving Triangle Parts ongruent to Measure istance Got It? Given: O, M is the midpoint of Prove: lm OlM 13. Use the flow chart to complete the proof. M Given Given Reflexive Property of M is the midpoint of. M M efinition of midpoint M M SSS Theorem nm nm orresponding parts of triangles are. /M /M Lesson heck o you know HOW? Name the postulate or theorem that you can use to show the triangles are congruent. Then explain why O M. 14. ircle the angles that are marked congruent. / /T /M /T /MT 15. ircle the sides that are marked congruent. T MT M T 16. ircle the sides that are congruent by the Reflexive Property of ongruence. T and MT and M T and T 17. Underline the correct postulate or theorem to complete the sentence. nt > nmt by SS / S / S / SSS. 18. Now explain why > M. orresponding parts of congruent triangles are congruent. T M opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. hapter 4 104

16 Lesson heck o you UNRSTN? rror nalysis ind and correct the error(s) in the proof. Given: KH > NH, /L > /M Prove: H is the midpoint of LM. L Proof: KH > NH because it is given. /L > /M because it is given. /KHL > /NHM because vertical angles are congruent. So, nkhl > nmhn by S Postulate. Since corresponding parts of congruent triangles are congruent, LH > MH. y the definition of midpoint, H is the midpoint of LM. K H N M Place a in the box if the statement is correct. Place an if it is incorrect. 19. /KHL > /NHM because vertical angels are congruent. 20. nkhl > nmhn by S Postulate. Underline the correct word to complete each sentence. 21. When you name congruent triangles, you must name corresponding vertices in a different / the same order. 22. To use the S Postulate, you need two pairs of congruent angles and a pair of included / nonincluded congruent sides. 23. To use the S Theorem, you need two pairs of congruent angles and a pair of included / nonincluded congruent sides. 24. Identify the error(s) in the proof. kkhl is not congruent to kmhn. The S Postulate cannot be used to opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. prove the triangles congruent. 25. orrect the error(s) in the proof. kkhl OkNHM by the S Theorem. Math Success heck off the vocabulary words that you understand. congruent corresponding proof Rate how well you can use congruent triangles. Need to review Now I get it! 105 Lesson 4-4

17 4-5 Isosceles and quilateral Triangles Vocabulary Review Underline the correct word to complete each sentence. 1. n equilateral triangle has two/ three congruent sides. 2. n equilateral triangle has acute / obtuse angles. 3. ircle the equilateral triangle Vocabulary uilder isosceles (adjective) eye SHS uh leez isosceles Related Words: equilateral, scalene efinition triangle is isosceles if it has two congruent sides. Main Idea: The angles and sides of isosceles triangles have special relationships. Use Your Vocabulary 4. Use the triangles below. Write the letter of each triangle in the correct circle(s) at the right. quilateral Isosceles Right,,,, opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. hapter 4 106

18 Theorems 4 3, 4 4, 4 5 Theorem 4-3 Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Theorem 4-4 onverse of Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Theorem 4-5 If a line bisects the vertex angle of an isosceles triangle, then the line is also the perpendicular bisector of the base. Vertex Vertex ngle ngle Legs Legs ase ase ase ase ngles ngles 5. If PQ > RQ in npqr, then / P > / R. 6. Underline the correct theorem number to complete the sentence. The theorem illustrated below is Theorem 4 3 / 4 4 / 4 5. If... Then... opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. Problem 1 Using the Isosceles Triangle Theorems T Got It? Is lwvs congruent to ls? Is TR congruent to TS? xplain. U W 7. The markings show that WV > WS. 8. Is /WVS > /S? xplain. R V S Yes. xplanations may vary. Sample: Since WV O WS, lwvs OlS by the Isosceles Triangle Theorem. 9. Is/R > /S? xplain. Yes. xplanations may vary. Sample: lr OlWVS and lwvs OlS, so lr OlS by the Transitive Property of ongruence. 10. Is TR > TS? xplain. Yes. xplanations may vary. Sample: Since lr OlS, TR O TS by the onverse of the Isosceles Triangle Theorem. 107 Lesson 4-5

19 Problem 2 Using lgebra Got It? Suppose ml What is the value of x? 11. Since >, n is isosceles. 12. Since n is isosceles, m/ 5 m/ x 13. Since bisects the vertex of an isosceles triangle, ' and m/ Use the justifications below to find the value of x. m/ 1 m/ 1x Triangle ngle-sum Theorem x Substitute x Simplify. x 5 63 Subtract 117 from each side. orollaries to Theorems 4 3 and 4 4 orollary to Theorem 4-3 If a triangle is equilateral, then the triangle is equiangular. orollary to Theorem 4-4 If a triangle is equiangular, then the triangle is equilateral. 15. Underline the correct number to complete the sentence. The corollary illustrated below is orollary to Theorem 4-3 / 4-4. If... Y Then... Problem 3 X Z inding ngle Measures Got It? Suppose the triangles at the right are isosceles triangles, where l, l, and l are vertex angles. If the vertex angles each have a measure of 58, what are ml and ml? 16. Which triangles are congruent by the Side-ngle-Side Theorem? k, k, k 17. Which angles are congruent by the Isosceles Triangle Theorem? l, l, l, l, l, l X Y Z opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. hapter 4 108

20 18. y the Triangle ngle-sum Theorem, m/ m/ Solve for m/. 2ml ml ml Since l > /, m/ Using the ngle ddition Postulate, m/ m/ Lesson heck o you UNRSTN? What is the relationship between sides and angles for each type of triangle? isosceles equilateral omplete. 22. n isosceles triangle has 2 congruent sides. 23. n equilateral triangle has 3 congruent sides. omplete each statement with congruent, isosceles, or equilateral. 24. The Isosceles Triangle Theorem states that the angles opposite the congruent sides are 9. congruent opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. 25. quilateral triangles are also 9 triangles. 26. The sides and angles of an 9 triangle are 9. Math Success heck off the vocabulary words that you understand. corollary legs of an isosceles triangle base of an isosceles triangle Need to review vertex angle of an isosceles triangle Rate how well you understand isosceles and equilateral triangles Now I get it! isosceles equilateral congruent base angles of an isosceles triangle 109 Lesson 4-5

21 4-6 ongruence in Right Triangles Vocabulary Review Write T for true or for false. T 1. Segments that are congruent have the same length. 2. Polygons that are congruent have the same shape but are not always the same size. T 3. In congruent figures, corresponding angles have the same measure. Vocabulary uilder hypotenuse (noun) hy PH tuh noos Related Word: leg leg hypotenuse leg efinition: The hypotenuse is the side opposite the right angle in a right triangle. Main Idea: The hypotenuse is the longest side in a right triangle. Use Your Vocabulary Underline the correct word(s) to complete each sentence. 4. One side of a right triangle is / is not a hypotenuse. 5. right triangle has one / two / three legs. 6. The length of the hypotenuse is always equal to / greater than / less than the lengths of the legs. Use the triangles at the right for xercises 7 and ross out the side that is NOT a hypotenuse. GH 8. ircle the leg(s). HI I G 2 3 H 4 opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. hapter 4 110

22 You can prove that two triangles are congruent without having to show that all corresponding parts are congruent. In this lesson, you will prove right triangles congruent by using one pair of right angles, a pair of hypotenuses, and a pair of legs. Theorem 4-6 Hypotenuse-Leg (HL) Theorem and onditions Theorem If... Then... If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent. npqr and nxyz are right triangles, PR > XZ, and PQ > XY P Q 9. To use the HL Theorem, the triangles must meet three conditions. omplete each sentence with right or congruent. R X Y Z npqr > nxyz There are two 9 triangles. The triangles have 9 hypotenuses. There is one pair of 9 legs. right congruent congruent opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. Use the information in the Take Note for xercises How do the triangles in the Take Note meet the first condition in xercise 9? xplain. lpqr and lxyz are right angles. 11. How do the triangles in the Take Note meet the second condition in xercise 9? xplain. The hypotenuses PR and XZ are marked congruent on the diagram. 12. How do the triangles in the Take Note meet the third condition in xercise 9? xplain. Legs PQ and XY are marked congruent on the diagram. 111 Lesson 4-6

23 Problem 1 Using the HL Theorem Got It? Given: /PRS and /RPQ are right angles, SP O QR Prove: kprs OkRPQ P Q 13. omplete each step of the proof. S R Given Given Reflexive Prop. of > /PRS and / RPQ are right angles. SP > QR PR > RP efinition of right triangle nprs and n RPQ are right triangles. HL Theorem nprs > n RPQ Problem 2 Writing a Proof Using the HL Theorem Got It? Given: O, is the perpendicular bisector of Prove: k Ok 14. ircle what you know because is the perpendicular bisector of. / and / are right angles. / and / are acute angles. is the midpoint of. is the midpoint of. 15. ircle the congruent legs. 16. Write the hypotenuse of each triangle. n n 17. omplete the proof. Statements Reasons 1) > 1) Given 2) / and / are right /s. 2) efinition of ' bisector 3) n and n are right ns. 3) efinition of right n 4) > 4) efinition of ' bisector 5) n > 5) HL Theorem opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. hapter 4 112

24 Lesson heck o you UNRSTN? rror nalysis Your classmate says that there is not enough information to determine whether the two triangles at the right are congruent. Is your classmate correct? xplain. Write T for true or for false. 18. There are three right angles. M L J K T 19. There are two right triangles. T T 20. There are two congruent hypotenuses. 21. There are no congruent legs. 22. You need to use the Reflexive Property of ongruence. 23. LJ > LJ is given. 24. o you always need three congruent corresponding parts to prove triangles congruent? xplain. xplanations may vary. Sample: No. You usually need three congruent corresponding parts, but not if the triangles are right triangles. opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. 25. Is your classmate correct? xplain. xplanations may vary. Sample: No. Since the hypotenuse and the shared leg of one right triangle are congruent to the hypotenuse and the shared leg ofanother right triangle, the triangles are congruent by the HL Theorem. Math Success heck off the vocabulary words that you understand. hypotenuse Need to review legs of a right triangle Rate how well you can use the Hypotenuse-Leg (HL) Theorem Now I get it! 113 Lesson 4-6

25 4-7 ongruence in Overlapping Triangles Vocabulary Review 1. ircle the common side of n and n. 2. ircle the common side of nxwz and nywz. WZ WX WY ZY 3. ircle the common side of nrst and nrpt. RP RS RT ST Vocabulary uilder overlapping (adjective) oh vur LP ing Other Word orm: overlap (noun) efinition: Overlapping events or figures have parts in common. Math Usage: Two or more figures with common regions are overlapping figures. Use Your Vocabulary ircle the common regions of the overlapping figures in the diagram at the right. 4. ng and n ng n nh ngh 5. n and nh n ngh ng nh 6. n and ng ng n ngh nh 7. n and ngh ng n ngh nh G H opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. hapter 4 114

26 Problem 1 Identifying ommon Parts Got It? What is the common side in k and k? 8. Separate and redraw n and n. 9. You drew twice, so the common side is. Problem 2 Using ommon Parts Got It? Given: k > k Prove: > 10. Use the information in the problem to complete the problem-solving model below. Know Need Plan k Ok; k ommon parts of Show k Ok. and k are k and k. Then use corresponding overlapping triangles. parts to prove O. opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. 11. Use the justifications below to complete each statement. Statements Reasons 1) n > k 1) Given 2) > 2) orresponding parts of > triangles are >. 3) / > l 3) orresponding parts of > triangles are >. 4) l > / 4) Vertical angles are congruent. 5) k > n 5) ngle-ngle-side (S) Theorem 6) > 6) orresponding parts of > triangles are >. 12. How could you use the onverse of the Isosceles Triangle Theorem to prove >? nswers may vary. Sample: l and l are corresponding parts of congruent triangles, so l Ol. is opposite l and is opposite l, so O by the converse of Isosceles Triangle Theorem. 115 Lesson 4-7

27 Problem 3 Using Two Pairs of Triangles Got It? Given: PS O RS, /PSQ O /RSQ P Prove: kqpt OkQRT 13. Give the reason for each statement in the proof. Given Given Reflexive Property of Q S T R PS RS PSQ RSQ ST ST and QS QS SS PST RST and QPS QRS orresponding parts of triangles are PT RT orresponding parts of triangles are PQ RQ Reflexive Property of SSS QPT QRT QT QT Problem 4 Separating Overlapping Triangles Got It? Given: l Ol, l Ol Prove: O 14. ircle the angles that are vertical angles. / / / / / / 15. Mark the angles that you know are congruent in each pair of separated triangles below. 16. Which triangles are congruent by S? xplain. opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. k and k; l Ol, l Ol and O. hapter 4 116

28 17. Which triangles are congruent by S? xplain. k and k; l and l are vertical and congruent, O since they are corresponding parts of k and k, and l Ol. 18. How can you prove >? and are corresponding parts of the congruent triangles k and k, so they are congruent. Lesson heck o you UNRSTN? In the figure at the right, which pair of triangles could you prove congruent first in order to prove that k Ok? xplain. 19. Is the hypotenuse of n congruent to the hypotenuse of n? xplain. xplanations may vary. Sample: Yes, by the Segment ddition Postulate 20. What else do you need to prove right angles congruent using HL? You need to show O or O. opyright by Pearson ducation, Inc. or its affiliates. ll Rights Reserved. 21. Which triangles can you prove congruent to find this? xplain. xplanations may vary. Sample: Use the Vertical ngles Theorem and SS to prove k Ok or k Ok. Math Success heck off the vocabulary words that you understand. congruent corresponding overlapping Rate how well you can identify congruent overlapping triangles. Need to review Now I get it! 117 Lesson 4-7