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# Using Congruent Triangles

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1. Using Congruent Triangles Ch 4 Lesson 5

2. CPCTC • Corresponding parts of congruent triangles are congruent (CPCTC): Once two triangles are proven to be congruent than all corresponding parts of the two triangles are congruent

3. Example #1 • Given AB II CD, BC II DA • Prove AB ≅ CD • Copy the diagram with signs to show parallel sides

4. Statement AB II CD BC II DA <1 ≅ <4 <2 ≅ <3 BD ≅ BD ABC ≅ BDC AB ≅ CD Reason Given Given Alternating Angles Alternating Angles Reflexive prop. Def of ≅ triangles (ASA) Corresponding Sides of congruent triangles are congruent Example #1: Prove AB ≅ CD

5. Statement AB II CD BC II DA <1 ≅ <4 <2 ≅ <3 BD ≅ BD ABC ≅ BDC AB ≅ CD Reason Given Given Alternating Angles Alternating Angles Reflexive prop. Def of ≅ triangles (ASA) Corresponding Sides of congruent triangles are congruent Example #1: Prove AB ≅ CD

6. Statement AB II CD BC II DA <1 ≅ <4 <2 ≅ <3 BD ≅ BD ABC ≅ BDC AB ≅ CD Reason Given Given Alternating Angles Alternating Angles Reflexive prop. Def of ≅ triangles (ASA) Corresponding Sides of congruent triangles are congruent Example #1: Prove AB ≅ CD

7. Statement AB II CD BC II DA <1 ≅ <4 <2 ≅ <3 BD ≅ BD ABC ≅ BDC AB ≅ CD Reason Given Given Alternating Angles Alternating Angles Reflexive prop. Def of ≅ triangles (ASA) Corresponding Sides of congruent triangles are congruent Example #1: Prove AB ≅ CD

8. Statement AB II CD BC II DA <1 ≅ <4 <2 ≅ <3 BD ≅ BD ABC ≅ BDC AB ≅ CD Reason Given Given Alternating Angles Alternating Angles Reflexive prop. Def of ≅ triangles (ASA) Corresponding Sides of congruent triangles are congruent Example #1: Prove AB ≅ CD

9. Statement AB II CD BC II DA <1 ≅ <4 <2 ≅ <3 BD ≅ BD ΔABC ≅ ΔBDC AB ≅ CD Reason Given Given Alternating Angles Alternating Angles Reflexive prop. Def of ≅ triangles (ASA) CPCTC Example #1: Prove AB ≅ CD

10. Statement AB II CD BC II DA <1 ≅ <4 <2 ≅ <3 BD ≅ BD ΔABC ≅ ΔBDC AB ≅ CD Reason Given Given Alternating Angles Alternating Angles Reflexive prop. Def of ≅ triangles (ASA) CPCTC Example #1: Prove AB ≅ CD

11. Example #2 Given: A is midpoint of MT A is midpoint of SR Prove: MS II TR

12. Statement A is midpoint of MT A is midpoint of SR MA ≅ AT & SA ≅ AR <SAM ≅ <RAT Δ SAM ≅ Δ RAT <M≅ <T & <S≅ <R MS II TR Reason Given Given Def of ≅ Vertical <s SAS Theorem CPCTC (alternating <s) CPCTC (alternating <s) Given: A is midpoint of MT A is midpoint of SRProve: MS II TR

13. Statement A is midpoint of MT A is midpoint of SR MA ≅ AT & SA ≅ AR <SAM ≅ <RAT Δ SAM ≅ Δ RAT <M≅ <T & <S≅ <R MS II TR Reason Given Given Def of ≅ Vertical <s SAS Theorem CPCTC (alternating <s) CPCTC (alternating <s) Given: A is midpoint of MT A is midpoint of SRProve: MS II TR

14. Statement A is midpoint of MT A is midpoint of SR MA ≅ AT & SA ≅ AR <SAM ≅ <RAT Δ SAM ≅ Δ RAT <M≅ <T & <S≅ <R MS II TR Reason Given Given Def of ≅ Vertical <s SAS Theorem CPCTC (alternating <s) CPCTC (alternating <s) Given: A is midpoint of MT A is midpoint of SRProve: MS II TR

15. Statement A is midpoint of MT A is midpoint of SR MA ≅ AT & SA ≅ AR <SAM ≅ <RAT Δ SAM ≅ Δ RAT <M≅ <T & <S≅ <R MS II TR Reason Given Given Def of ≅ Vertical <s SAS Theorem CPCTC (alternating <s) CPCTC (alternating <s) Given: A is midpoint of MT A is midpoint of SRProve: MS II TR

16. Statement A is midpoint of MT A is midpoint of SR MA ≅ AT & SA ≅ AR <SAM ≅ <RAT Δ SAM ≅ Δ RAT <M≅ <T & <S≅ <R MS II TR Reason Given Given Def of ≅ Vertical <s SAS Theorem CPCTC (alternating <s) CPCTC (alternating <s) Given: A is midpoint of MT A is midpoint of SRProve: MS II TR

17. Statement A is midpoint of MT A is midpoint of SR MA ≅ AT & SA ≅ AR <SAM ≅ <RAT Δ SAM ≅ Δ RAT <M≅ <T & <S≅ <R MS II TR Reason Given Given Def of ≅ Vertical <s SAS Theorem CPCTC (alternating <s) CPCTC (alternating <s) Given: A is midpoint of MT A is midpoint of SRProve: MS II TR

18. Example #3 • Given: <1≅ <2 & <3 ≅ <4 • Prove ΔBCE ≅ ΔDCE

19. Statement <1≅ <2 & <3 ≅ <4 CA ≅CA ΔACD ≅ ΔABC CB ≅ CB In ΔDEC & ΔBEC <1 ≅<2 CD ≅CB CE ≅CE ΔBCE ≅ ΔDCE Reason Given Reflexive ASA CPCTC Given CPCTC Reflexive SAS Given: <1≅ <2 & <3 ≅ <4Prove: ΔBCE ≅ ΔDCE

20. Statement <1≅ <2 & <3 ≅ <4 CA ≅CA ΔACD ≅ ΔABC CB ≅ CB In ΔDEC & ΔBEC <1 ≅<2 CD ≅CB CE ≅CE ΔBCE ≅ ΔDCE Reason Given Reflexive ASA CPCTC Given CPCTC Reflexive SAS Given: <1≅ <2 & <3 ≅ <4Prove: ΔBCE ≅ ΔDCE

21. Statement <1≅ <2 & <3 ≅ <4 CA ≅CA ΔACD ≅ ΔABC CB ≅ CB In ΔDEC & ΔBEC <1 ≅<2 CD ≅CB CE ≅CE ΔBCE ≅ ΔDCE Reason Given Reflexive ASA CPCTC Given CPCTC Reflexive SAS Given: <1≅ <2 & <3 ≅ <4Prove: ΔBCE ≅ ΔDCE

22. Statement <1≅ <2 & <3 ≅ <4 CA ≅CA ΔACD ≅ ΔABC CB ≅ CB In ΔDEC & ΔBEC <1 ≅<2 CD ≅CB CE ≅CE ΔBCE ≅ ΔDCE Reason Given Reflexive ASA CPCTC Given CPCTC Reflexive SAS Given: <1≅ <2 & <3 ≅ <4Prove: ΔBCE ≅ ΔDCE

23. Statement <1≅ <2 & <3 ≅ <4 CA ≅CA ΔACD ≅ ΔABC CB ≅ CB In ΔDEC & ΔBEC <1 ≅<2 CD ≅CB CE ≅CE ΔBCE ≅ ΔDCE Reason Given Reflexive ASA CPCTC Given CPCTC Reflexive SAS Given: <1≅ <2 & <3 ≅ <4Prove: ΔBCE ≅ ΔDCE

24. Statement <1≅ <2 & <3 ≅ <4 CA ≅CA ΔACD ≅ ΔABC CB ≅ CB In ΔDEC & ΔBEC <1 ≅<2 CD ≅CB CE ≅CE ΔBCE ≅ ΔDCE Reason Given Reflexive ASA CPCTC Given CPCTC Reflexive SAS Given: <1≅ <2 & <3 ≅ <4Prove: ΔBCE ≅ ΔDCE

25. Statement <1≅ <2 & <3 ≅ <4 CA ≅CA ΔACD ≅ ΔABC CB ≅ CB In ΔDEC & ΔBEC <1 ≅<2 CD ≅CB CE ≅CE ΔBCE ≅ ΔDCE Reason Given Reflexive ASA CPCTC Given CPCTC Reflexive SAS Given: <1≅ <2 & <3 ≅ <4Prove: ΔBCE ≅ ΔDCE

26. Statement <1≅ <2 & <3 ≅ <4 CA ≅CA ΔACD ≅ ΔABC CB ≅ CB In ΔDEC & ΔBEC <1 ≅<2 CD ≅CB CE ≅CE ΔBCE ≅ ΔDCE Reason Given Reflexive ASA CPCTC Given CPCTC Reflexive SAS Given: <1≅ <2 & <3 ≅ <4Prove: ΔBCE ≅ ΔDCE

27. Statement <1≅ <2 & <3 ≅ <4 CA ≅CA ΔACD ≅ ΔABC CB ≅ CB In ΔDEC & ΔBEC <1 ≅<2 CD ≅CB CE ≅CE ΔBCE ≅ ΔDCE Reason Given Reflexive ASA CPCTC Given CPCTC Reflexive SAS Given: <1≅ <2 & <3 ≅ <4Prove: ΔBCE ≅ ΔDCE

28. Statement <1≅ <2 & <3 ≅ <4 CA ≅CA ΔACD ≅ ΔABC CB ≅ CB In ΔDEC & ΔBEC <1 ≅<2 CD ≅CB CE ≅CE ΔBCE ≅ ΔDCE Reason Given Reflexive ASA CPCTC Given CPCTC Reflexive SAS Given: <1≅ <2 & <3 ≅ <4Prove: ΔBCE ≅ ΔDCE