1. DRILL Statements Reasons
Given: N is the midpoint of LW N is the midpoint of SK
Prove:
Statements Reasons
N is the midpoint of LW N is the midpoint of SK
Given
Definition of Midpoint
Vertical Angles are congruent
SAS Postulate

2. 8.2 Proving Triangles are Congruent: ASA and AAS
Geometry
Mr. Calise

3. 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.

4. Postulate 21: Angle-Side-Angle (ASA) Congruence Postulate
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.

5. Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem
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.

6. Third Angles Theorem
If two angles in one triangle are congruent to two angles in another triangle then the third angles must also be congruent.

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

8. Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem
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.

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

10. Ex. 1 Developing Proof
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.

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

12. Ex. 1 Developing Proof
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.

13. Ex. 1 Developing Proof
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. 1 2 3 4

14. Ex. 1 Developing Proof
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. 1 2 3 4

15. Ex. 2 Proving Triangles are Congruent
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.

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

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

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

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

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

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

22. 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.