1. 4.6 Prove Triangles Congruent by ASA and AAS
You will use two more methods to prove congruences.
Essential Question: If a side of one triangle is congruent to a side of another triangle, what information about the angles would allow you to prove the triangles are congruent?
You will learn how to answer this question by learning about the ASA Postulate and the AAS Theorem.

2. 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.

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

4. 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

5. 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
AAS; because they are vertical angles.
RTS UTV

6. 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°

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
ANSWER
The correct answer is B.

11. 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?
ANSWER
AAS Congruence Theorem.

12. Daily Homework Quiz
Tell whether each pair of triangle are congruent by SAS, ASA, SSS, AAS or HL. If it is not possible to prove the triangle congruent, write not necessarily congruent. 1.
ANSWER
ASA .

13. Daily Homework Quiz
Tell whether each pair of triangle are congruent by SAS, ASA, SSS, AAS or HL. If it is not possible to prove the triangle congruent, write not necessarily congruent. 2.
ANSWER
not necessarily congruent .

14. Daily Homework Quiz
Write flow proof.
Given : BD bisects ABC, A C
Prove : ABD CBD 3.

15. Daily Homework Quiz
ANSWER

16. You will use two more methods to prove congruences.
Essential Question: If a side of one triangle is congruent to a side of another triangle, what information about the angles would allow you to prove the triangles are congruent?
You will use two more methods to prove congruences.
• Triangles are congruent by the
ASA Congruence Postulate.
AAS Congruence Theorem.
• Another format for proofs is the
flow proof.
The triangles will be congruent if
the conditions of the ASA
Congruence Postulate or of the
AAS Congruence Theorem are met.