1.In VGB, which sides include B?

2.In STN, which angle is included between NS and TN?

3.Which triangles can you prove congruent?

Tell whether you would use the SSS or SAS Postulate.

4.What other information do you need to prove

DWO DWG?

5.Can you prove SED BUT from the information given? Explain.

N

Triangle Congruence by SSS and SASTriangle Congruence by SSS and SASGEOMETRY LESSON 4-2GEOMETRY LESSON 4-2

BG and BV

APB XPY; SAS

If you know DO DG, the triangles are by SSS; if you know DWO DWG, they are by SAS.

No; corresponding angles are not between corresponding sides.

4-2

In JHK, which side is included between the given pair of angles?

1. J and H 2. H and K

In NLM, which angle is included between the given pair of sides?

3. LN and LM 4. NM and LN

Give a reason to justify each statement.

5. PR PR 6. A D

(For help, go to Lesson 4-2.)

GEOMETRY LESSON 4-3GEOMETRY LESSON 4-3

Triangle Congruence by ASA and AASTriangle Congruence by ASA and AAS

4-3

Check Skills You’ll Need

JH HK

L N

By the Reflexive Property of Congruence, a segment is congruent to itself

Third Angles Theorem

Triangle Congruence by ASA and AASTriangle Congruence by ASA and AASGEOMETRY LESSON 4-3GEOMETRY LESSON 4-3

4-3

An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side.

Triangle Congruence by ASA and AASTriangle Congruence by ASA and AASGEOMETRY LESSON 4-3GEOMETRY LESSON 4-3

4-3

Triangle Congruence by ASA and AASTriangle Congruence by ASA and AASGEOMETRY LESSON 4-3GEOMETRY LESSON 4-3

4-3

You can use the Third Angles Theorem to prove another congruence relationship based on ASA. This theorem is Angle-Angle-Side (AAS).

Triangle Congruence by ASA and AASTriangle Congruence by ASA and AASGEOMETRY LESSON 4-3GEOMETRY LESSON 4-3

4-3

Suppose that F is congruent to C and I is not congruent to C. Name the triangles that are congruent by the ASA Postulate.

Triangle Congruence by ASA and AASTriangle Congruence by ASA and AASGEOMETRY LESSON 4-3GEOMETRY LESSON 4-3

Therefore, FNI CAT GDO by ASA.

If F C, then F C G

The diagram shows N A D and FN CA GD.

4-3

Quick Check

Using ASA

Write a paragraph proof.

Given: A B, AP BP

Prove: APX BPY

Triangle Congruence by ASA and AASTriangle Congruence by ASA and AASGEOMETRY LESSON 4-3GEOMETRY LESSON 4-3

It is given that A B and AP BP.

APX BPY by the Vertical Angles Theorem.

Because two pairs of corresponding angles and their included sides are congruent, APX BPY by ASA.

4-3

Quick Check

Writing a proof using ASA

Write a Plan for Proof that uses AAS.

Given: B D, AB || CD

Prove: ABC CDA

Triangle Congruence by ASA and AASTriangle Congruence by ASA and AASGEOMETRY LESSON 4-3GEOMETRY LESSON 4-3

By the Reflexive Property, AC AC so ABC CDA by AAS.

Then ABC CDA if a pair of corresponding sides are congruent.

Because AB || CD, BAC DCA by the Alternate Interior Angles Theorem.

4-3

Quick Check

Planning a Proof using AAS

Write a two-column proof that uses AAS.

Given: B D, AB || CD

Prove: ABC CDA

Statements Reasons

Triangle Congruence by ASA and AASTriangle Congruence by ASA and AASGEOMETRY LESSON 4-3GEOMETRY LESSON 4-3

1. B D, AB || CD 1. Given

5. ABC CDA 5. AAS Theorem

3. BAC DCA 3. Alternate Interior Angle Theorem .

4-3

4. AC CA 4. Reflexive Property of Congruence

Quick Check

Writing a proof using AAS

2. BAC & DCA are AIA 2. Definition of Alternate Interior Angle.

1. Which side is included between R and F in FTR?

2. Which angles in STU include US?

Tell whether you can prove the triangles congruent by ASA or AAS. If you can, state a triangle congruence and the postulate or theorem you used. If not, write not possible.

3. 4. 5.

RF

S and U

Triangle Congruence by ASA and AASTriangle Congruence by ASA and AASGEOMETRY LESSON 4-3GEOMETRY LESSON 4-3

GHI PQRAAS

not possible ABX ACXAAS

4-3