Other methods of Other methods of Proving Triangles Proving Triangles

Congruent Congruent (AAS), (HL)(AAS), (HL)

4-54-5

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

GUIDED PRACTICE for Examples 1 and 2

SOLUTION

1.

GivenS U

The vertical angles are congruent

RTS UTV

GivenRS UV

STATEMENTS REASONS

In the diagram at the right, what postulate or theorem can you use to prove that RST VUT ? Explain.

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

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°1m 2m m+ +

2. Angle Addition Postulate and definition of straight angle

2. 4m 2m 5m+ + = 180°

3. Alternate Interior Angles Theorem

3. 1 4 , 3 5

5. Substitution Property of Equality

5. 1m 2m 3m+ + = 180°

4. Definition of congruent angles

4. 1m = 4m 3m = 5m,

STATEMENTS REASONS

GIVEN ABC

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

EXAMPLE 4 Standardized Test Practice

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.

EXAMPLE 4 Standardized Test Practice

The correct answer is B.

ANSWER

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?

GivenABE ADE

Both are right angle triangle.

Definition of right triangle

AEB AED

Reflexive Property of CongruenceBD DB

STATEMENTS REASONS

AAS Congruence TheoremABE ADE

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

SOLUTIONProved 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.

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