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I Using Corresponding Parts of Congruent Triangles What You'll Learn @ Check Skills You'll Need • To identify congruent overlapping triangles for Help '~ D A KDz - T R~S J~N Some triangle relationships are difficult to see because the triangles overlap. Overlapping triangles may have a common side or angle. You can simplify your work with overlapping triangles by separating and redrawing the triangles. Overlapping triangles share part or all of one or more sides. A Identifying Separate and redraw L.DFG and L.EHG. Identify the common angle. I • 2. Can you conclude that the triangles are congruent? Explain. a. L.AZK and L.DRS b. L.SDR and L.JTN c. L.ZKA and L.NJT To identify overlapping triangles in scaffolding, as in Example 1 I<_~E 1 1 and 4-3 the next two figures in this pattern have? ... And Why :1><:1: I, I 1. How many triangles will • To prove two triangles congruent by first proving two other triangles congruent @ Quick Check I, o Common Parts G D Common angle E D H~E The diagram at the left shows triangles from the scaffolding that workers used when they repaired and cleaned the Statue of Liberty. a. Name the common side in MDC and L.BCD. b. Name another pair of triangles that share a common side. Name the common side. Engineering In overlapping triangles, a common side or angle is congruent to itself by the Reflexive Property of Congruence. Lesson 4-7 Using Corresponding Parts of Congruent Triangles 241 Proof ___ ..~ _ Using Common Parts "~ Given: LZXW == L YWX, LZWX == Z L YXW Write a plan and then a proof to show that the -two "outside" segments are congruent. Prove: ZW == Y X First, separate the -overlapping - triangles. ZW == YX by CPCTCif L,ZXW == L,ywx. Show this congruence by ASA. Plan: y w~x Z y w~x w~x Proof: I LZXW==LYWX Given I~ I WX == WX • ReflexiveProp. of == 1 ~ I LZWX==Lyxwl/ -I L,-Z-XW-==-L,-Y1-VX-1 ASA / • Postulate I ZW == YX 1 CPCTC Given @ Quick Check 8 Write a plan and then a proof. Given: Prove: A B c D -L,ACD -DE L,BDC CE == == Sometimes you can prove one pair of triangles congruent and then use their congruent corresponding parts to prove another pair congruent. Proof Using Two Pairs of Triangles ~ Given: Prove: In the quilt, E is the midpoint of AC and DB. L,GED == L,JEB Write a plan and then a proof. Plan: L,GED == L,JEB by ASA if LD == LB. These angles are congruent by CPCTC if MED == L,CEB. These triangles are congruent by SAS. E is the midpoint of AC and DB, so AE == CE and DE == BE. LAED == LCEB because vertical angles are congruent. Therefore, MED == L,CEB by SAS. LD == LB by CPCTC, and LGED == LJEB because they are vertical angles. Therefore, L,GED == L,JEB by ASA. Proof: II @ Quick Check e Write a plan and then a proof. Given: PS Prove: L,QPT == RS, LPSQ == == Q p R LRSQ L,QRT s 242 Chapter 4 Congruent Triangles •• When triangles overlap, you can keep track of information by drawing other diagrams that separate the overlapping triangles. Separating Overlapping Triangles Given: CA::= CE, BA ::= c DE Write a plan and then a proof to show that two small segments inside -the triangle are congruent. Prove: BX::= DX - Plan: BX ::= DX by CPCTC if f:oBXA ::= f:oDXE. This congruence holds by AAS if LABX ::= LEDX. These are congruent by CPCTC in f:oBAE and f:oDEA, which are congruent by SAS. A E B Proof: A~E Statements \t' Connection --~ Japanese paper-folding J""" ::If origami involves many :dapping triangles. @QuickCheck 4) A~E Reasons 1. BA ::= DE 1. Given 2. CA ::= CE 2. Given 3. LCAE::= LCEA 3. Isosceles Triangle Theorem 4. AE 4. Reflexive Property of Congruence 5. SAS ::= AE 5. f:oBAE ::= f:oDEA .:.aI-World D 6. LABE ::= LEDA 6. CPCTC 7. LBXA ::= LDXE 8. f:oBXA ::= f:oDXE 7. Vertical angles are congruent. 8. AAS 9. BX ::= DX 9.CPCTC 0 Plan a proof. Separate the overlapping c triangles in your plan. Then follow your plan and write a proof. LCAD::= LEAD,LC::= Prove: BD::= FD Given: LE A E EXERCISES For more exercises, see Extra Skill, Word Problem, and Proof Practice. 11'I~,{fJl " Practice by Example Example 1 In each diagram, the red and blue triangles are congruent. Identify their common side or angle. (page 241) 2. ~for Help '-V )~{ M lesson 4-7 Using Corresponding E 3. ybz x~'[ Parts of Congruent Triangles 243 Separate and redraw the indicated triangles. Identify any common angles or sides. 6. 6.JKL and 6.MLK 5. 6.ACB and 6.PRB 4. 6.PQS and 6.QPR P s Example 2 B R 7. Developing Proof Complete (page 242) J6M AvzP [g]Q P the flow proof. Given: L T ~ LR, PQ ~ PV Prove: LPQT ~ LPVR TV- I ILT~LRI~ a.~ LTPQ~ LRPV b.~ PQ~W ~ --- I ). I 1/ ~R / 6.TPQ~ 6.RPVI ). I LPQT~ LPVR I e.l d.~ c.~ -Proof Write a plan and then a proof. 8. Given: RS ~ UT, RT ~ US Prove: 6.RST ~ 6. UTS S 9. Given: QD ~ UA,LQDA~LUAD Prove: 6.QDA ~ 6.UAD Q T ROU w Examples 3, 4 (pages 242 and 243) 10. Given: Ll ~ U ~ D A V L2, L3 ~ L4 Prove: 6.QET ~ 6.QEU T 11. Given: AD ~ ED, D is the midpoint of BF. Prove: 6.ADC ~ 6.EDG A B Q U o E Apply Your Skills Open-Ended Draw the diagram described. 12. Draw a vertical segment on your paper. On the right side of the segment draw two triangles that share the given segment as a common side. 13. Draw two triangles that have a common angle. ~.nline Homework Video Tutor Visit: PHSchool.com Web Code: aue-0407 244 ------ Chapter 4 14. Draw two regular pentagons, each with its five diagonals. a. In one, shade two triangles that share a common angle. b. In the other, shade two triangles that share a common side. 15. Draw two regular hexagons and their diagonals. For these diagrams, do parts (a) and (b) of the preceding exercise. Congruent Triangles