1. Other methods of Proving Triangles Congruent (AAS), (HL)
4-5

2. 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

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

4. GUIDED PRACTICE
for Examples 1 and 2
ANSWER
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

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

6. 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

7. EXAMPLE 4
Standardized Test Practice

8. 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.

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

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

11. GUIDED PRACTICE
for Examples 3 and 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.

12. GUIDED PRACTICE
for Examples 3 and 4
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.