4.6 prove triangles congruent by asa and aas you will use two more methods to prove congruences....

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.

Warm-Up ExercisesEXAMPLE 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.

Warm-Up ExercisesEXAMPLE 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.

Warm-Up ExercisesEXAMPLE 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

Warm-Up ExercisesGUIDED 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.

AAS; because they are vertical angles.

RTS UTV


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

Warm-Up ExercisesEXAMPLE 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

Warm-Up ExercisesEXAMPLE 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.


The correct answer is B.

ANSWER


In Example 3, suppose ABE ADE is also given. What theorem or postulate besides ASA can you use to prove that

3.

ABE ADE?

ANSWER

AAS Congruence Theorem.

Warm-Up ExercisesDaily 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.

ANSWER ASA .

1.


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 .


Write flow proof.Given : BD bisects ABC, A CProve : ABD CBD

3.


ANSWER

• 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?

• Triangles are congruent by theASA Congruence Postulate.• Triangles are congruent by theAAS Congruence Theorem.• Another format for proofs is theflow proof.

The triangles will be congruent ifthe conditions of the ASACongruence Postulate or of theAAS Congruence Theorem are met.

4.6 prove triangles congruent by asa and aas you will use two more methods to prove congruences....

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