1. EXAMPLE 1 Identify congruent triangles Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use. SOLUTION The vertical angles are congruent, so two pairs of angles and a pair of non-included sides are congruent. The triangles are congruent by the AAS Congruence Theorem. a.

2. EXAMPLE 1 Identify congruent triangles b. There is not enough information to prove the triangles are congruent, because no sides are known to be congruent. c. Two pairs of angles and their included sides are congruent. The triangles are congruent by the ASA Congruence Postulate.

3. EXAMPLE 2 Prove the AAS Congruence Theorem Prove the Angle-Angle-Side Congruence Theorem. Write a proof. GIVEN BC EF A D, C F, PROVE ABC DEF

4. GUIDED PRACTICE for Examples 1 and 2 SOLUTION 1. Given S U The vertical angles are congruent RTS UTV Given RS UV STATEMENTS REASONS In the diagram at the right, what postulate or theorem can you use to prove that RST VUT ? Explain.

5. GUIDED PRACTICE for Examples 1 and 2 Therefore are congruent because vertical angles are congruent so two pairs of angles and a pair of non included side are congruent. The triangle are congruent by AAS Congruence Theorem. RTS UTV ANSWER

6. GUIDED PRACTICE for Examples 1 and 2 2. Rewrite the proof of the Triangle Sum Theorem on page 219 as a flow proof. 1. Draw BD parallel to AC. 1. Parallel Postulate PROVE 3 = 180° 1m2mm++ 2. Angle Addition Postulate and definition of straight angle 2.4m2m5m++ = 180° 3. Alternate Interior Angles Theorem 3.1 4, 3 5 5. Substitution Property of Equality 5. 1m2m3m++ = 180° 4. Definition of congruent angles 4.1m= 4m 3m= 5m, STATEMENTS REASONS GIVEN ABC

7. EXAMPLE 3 Write a flow proof In the diagram, CE BD and  CAB CAD. Write a flow proof to show ABE ADE GIVEN CE BD,  CAB CAD PROVE ABE ADE

8. EXAMPLE 4 Standardized Test Practice

9. EXAMPLE 4 Standardized Test Practice The locations of tower A, tower B, and the fire form a triangle. The dispatcher knows the distance from tower A to tower B and the measures of A and B. So, the measures of two angles and an included side of the triangle are known. By the ASA Congruence Postulate, all triangles with these measures are congruent. So, the triangle formed is unique and the fire location is given by the third vertex. Two lookouts are needed to locate the fire.

10. EXAMPLE 4 Standardized Test Practice The correct answer is B. ANSWER

11. GUIDED PRACTICE for Examples 3 and 4 SOLUTION In Example 3, suppose ABE ADE is also given. What theorem or postulate besides ASA can you use to prove that 3. ABE ADE ? Given ABEADE Both are right angle triangle. Definition of right triangle AEBAED Reflexive Property of Congruence BD DB STATEMENTS REASONS AAS Congruence Theorem ABE ADE

12. GUIDED PRACTICE for Examples 3 and 4 4. What If? In Example 4, suppose a fire occurs directly between tower B and tower C. Could towers B and C be used to locate the fire? Explain SOLUTION Proved by ASA congruence The locations of tower B, tower C, and the fire form a triangle. The dispatcher knows the distance from tower B to tower C and the measures of B and C. So, he knows the measures of two angles and an included side of the triangle.

13. By the ASA Congruence Postulate, all triangles with these measures are congruent. No triangle is formed by the location of the fire and tower, so the fire could be anywhere between tower B and C. GUIDED PRACTICE for Examples 3 and 4