4.4 Proving Triangles are Congruent: ASA and AAS

1. 4.4 Proving Triangles are Congruent: ASA and AAS Geometry Mrs. Spitz Fall 2004

2. Objectives: • Prove that triangles are congruent using the ASA Congruence Postulate and the AAS Congruence Theorem • Use congruence postulates and theorems in real-life problems.

3. Assignment • 4.4 pp. 223-225 #1-22 all • Quiz after this section

4. If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent. Postulate 21: Angle-Side-Angle (ASA) Congruence Postulate

5. If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent. Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem

6. Given: A  D, C  F, BC  EF Prove: ∆ABC  ∆DEF Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem

7. You are given that two angles of ∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is, B  E. Notice that BC is the side included between B and C, and EF is the side included between E and F. You can apply the ASA Congruence Postulate to conclude that ∆ABC  ∆DEF. Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem

8. Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. Ex. 1 Developing Proof

9. A. In addition to the angles and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. You can use the AAS Congruence Theorem to prove that ∆EFG  ∆JHG. Ex. 1 Developing Proof

10. Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. Ex. 1 Developing Proof

11. B. In addition to the congruent segments that are marked, NP  NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent. Ex. 1 Developing Proof

12. Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. UZ ║WX AND UW ║WX. Ex. 1 Developing Proof 1 2 3 4

13. The two pairs of parallel sides can be used to show 1  3 and 2  4. Because the included side WZ is congruent to itself, ∆WUZ  ∆ZXW by the ASA Congruence Postulate. Ex. 1 Developing Proof 1 2 3 4

14. Given: AD ║EC, BD  BC Prove: ∆ABD  ∆EBC Plan for proof: Notice that ABD and EBC are congruent. You are given that BD  BC . Use the fact that AD ║EC to identify a pair of congruent angles. Ex. 2 Proving Triangles are Congruent

15. Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: 1. Proof:

16. Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: 1. Given Proof:

17. Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given Given Proof:

18. Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given Given Alternate Interior Angles Proof:

19. Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given Given Alternate Interior Angles Vertical Angles Theorem Proof:

20. Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given Given Alternate Interior Angles Vertical Angles Theorem ASA Congruence Theorem Proof:

21. Note: • You can often use more than one method to prove a statement. In Example 2, you can use the parallel segments to show that D  C and A  E. Then you can use the AAS Congruence Theorem to prove that the triangles are congruent.