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4.5 HL and overlapping triangles ink.notebook November 14, 2016 Page 168 Page 167 Page 166 4.5 HL and Overlapping Triangles Lesson Objectives Standards Lesson Notes 4.5 HL and Overlapping Triangles Press the tabs to view details. Lesson Objectives Standards Lesson Notes After this lesson, you should be able to successfully use HL to prove triangles are congruent. You will also learn how to prove overlapping triangles are congruent. Press the tabs to view details. 1 4.5 HL and overlapping triangles ink.notebook Lesson Objectives Standards Lesson Notes G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. November 14, 2016 HYPOTENUSELEG (HL) CONGRUENCE THEOREM If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second triangle, then the two triangles are ______________. B E G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. G.CO.10 Prove theorems about triangles. A C D F For each diagram, determine which pairs of triangles can be proved congruent by the HL Theorem. a) b) c) 2 4.5 HL and overlapping triangles ink.notebook November 14, 2016 E 1. Given: AC ¤ EC, AB ¼ BD, ED ¼ BD, and AC is a bisector of BD A Prove: ÆABC ¤ ÆEDC B Statements D C Reasons 1. AC ¤ EC,AB ¼ BD, ED ¼ BD, 1. and AC is a bisector of BD 2. ÚB and ÚD are 2. Defn of _____________ 3. ÆABC and ÆEDC are 3. Defn of _____________ 4. BC ¤ DC 4. Defn of _____________ 5. ÆABC ¤ ÆEDC 5. Sometimes when triangles overlap each other, they share corresponding parts. When trying to picture the corresponding parts, try redrawing the triangles separately. B C 30¡ A D B A C 30¡ D A D 2. In the figure ΔABD ¤ ΔDCA and mÚBDA = 30°. Find mÚCAD = _____ and mÚCDA = ______ 3 4.5 HL and overlapping triangles ink.notebook November 14, 2016 3. In the figure ΔABC ¤ ΔDCB, mÚABC = 110, and mÚD = 20. Find mÚACB = _______ and mÚDCB = ________ D A B C 4. Given: ÚA and ÚB are rt Ús, ÚACD ¤ ÚBDC, AD ¤ BC, AC ¤ BD A B Prove: ÆADC ¤ ÆBCD D Statements 30 C Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 4 4.5 HL and overlapping triangles ink.notebook November 14, 2016 PRACTICE On the Worksheet Using Congruent Triangles CPCTC 1. R P A 2. 1 B C D S Y A 2 E 5 4.5 HL and overlapping triangles ink.notebook Tell whether the HL Theorem can be applied to prove the triangles congruent. If possible, write the triangle congruence. 3. B 4. A R G Tell whether the HL Theorem can be applied to prove the triangles congruent. If possible, write the triangle congruence. U 5. T M F D Z C O L K 8. H 6. T E H F N W Tell whether the HL Theorem can be applied to prove the triangles congruent. If possible, write the triangle congruence. 7. November 14, 2016 Y 9. Given: ÚYEB & ÚYET are rt angles and ÚYBE ¤ ÚYTE Prove: èBYE ¤ èTYE I B N K H E T J A 6 4.5 HL and overlapping triangles ink.notebook State if the two triangles are congruent. If they are, state how you know. 10. 11. November 14, 2016 State if the two triangles are congruent. If they are, state how you know. 12. 13. State if the two triangles are congruent. If they are, state how you know. 14. 15. Missing Parts 7 4.5 HL and overlapping triangles ink.notebook November 14, 2016 16. Given: Ú1 ¤ Ú2 Prove: èABD ¤ èCBD By SAS Name the Part or Parts needed to prove the è’s ¤ by the given è¤ theorem. You may mark the reflexive sides or angles without stating them. A D 1 2 B C Need: _______________________ 17. Given: ÚA ¤ ÚC Prove: èEAB ¤ èDC By SAS E A 18. Given: Ú1 ¤ Ú2 Prove: èDAB ¤ èDCB By AAS D B C D A 12 B C Need: _______________________ & _______________________ Need: _______________________ 8 4.5 HL and overlapping triangles ink.notebook November 14, 2016 20. 19. A 1 B C 2 Prove: èABD ¤ èBAC By SSS Hint: Draw separate è’s B A If needed. E D D C E Need: _______________________ Need: _______________________ & _______________________ 21. Given: NO = NM Prove: èMNQ ¤ èONP By SSS M Q N O 22. Given: AB = BC Prove: èABD ¤ èCBD By HL A B 1 2 C D P Need: _______________________ Need: _______________________ & _______________________ & _______________________ 9 4.5 HL and overlapping triangles ink.notebook November 14, 2016 24. Given: Ú1 ¤ Ú2 Prove: èABD ¤ èCBD By ASA 23. Given: ÚB & ÚD are right Ú’s Prove: èADC ¤ èCBD By HL A D 34 B D C A Need: _______________________ (2 possible answers) D 25. 5 C Need: _______________________ 26. Given: èPQR ¤ èSQV Name all pairs of corresponding sides and angles. 34 6 12 B P Q V R 5 A 1 2 B 6 C _____ ¤ _____ Ú_____ ¤ Ú _____ S _____ ¤ _____ Ú_____ ¤ Ú _____ Need: _______________________ _____ ¤ _____ Ú_____ ¤ Ú _____ & _______________________ 10 4.5 HL and overlapping triangles ink.notebook November 14, 2016 Answers: 1. Given, ÚPSY ¤ ÚASR, Vert Ú Thm, èPSY ¤ èASR, AAS, CPCTC 3. Yes, èABD ¤ èDCA 5. Yes, èMNU ¤ èMNL 7. Yes, èOKN ¤ èAHN 9. Given, all rt Ú’s are ¤, reflexive, AAS 11. No 13. No 15. Yes, SAS 19. ÚD ¤ ÚE, Ú1 ¤ Ú2 25. Ú1 ¤ Ú2, Ú5 ¤ Ú6 11