1. Tell whether the pair of triangles is congruent or not and why.
ANSWER
Yes; HL Thm.

2. Proving triangles congruent.
Target
Proving triangles congruent.
GOAL:
4.6 Use sides and angles to prove triangles congruent.

3. Vocabulary
included side – the side between vertices of two angles
ASA (Angle-Side-Angle) 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 two triangles are congruent.
AAS (Angle-Angle-Side) Congruence Theorem
If two angles and the non-included side of one
triangle are congruent to two angles and the non-
included side of a second triangle,
then the two triangles are congruent.

4. 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.

5. EXAMPLE 1
Identify congruent triangles
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.
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.

6. Flow proof 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
Flow proof

7. 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
Vertical Angles Thm.
RTS UTV
RST VUT
AAS Congruence Thm.

8. 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

9. EXAMPLE 4
Standardized Test Practice

10. 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.

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

12. 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.
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
ANSWER
No triangle is formed by the location of the fire and towers, so the fire could be anywhere between towers B and C.