Lesson 4-5

Lesson 4-5

Proving Congruence:ASA and AAS

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Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove they are congruent, write not possible.

1. 2. 3.

4. 5.

6. If AB RS and BC ST, what additional congruence statement would be necessary to prove ABC RST by the SAS postulate?

Standardized Test Practice:

A CB DA R C T A T B S

5-Minute Check on Lesson 4-45-Minute Check on Lesson 4-45-Minute Check on Lesson 4-45-Minute Check on Lesson 4-4 Transparency 4-5

Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove they are congruent, write not possible.

1. 2. 3.

SAS SSS SSS

4. 5.

not possible SSS

6. If AB RS and BC ST, what additional congruence statement would be necessary to prove ABC RST by the SAS postulate?

Standardized Test Practice:

A CB DA R C T A T B S

Objectives

• Use the ASA Postulate to test for triangle congruence

• Use the AAS Theorem to test for triangle congruence

Vocabulary

• Included side – the side in common between two angles (end points are the vertexes)

Postulates and Theorems

• Angle-Side-Angle (ASA) Postulate: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

• Angle-Angle-Side (AAS) Theorem: If two angles and a non-included side of one triangle are congruent to the corresponding two angles and side of another triangle, then the triangles are congruent.

Angle – Side – Angle (ASA)

Statements Reasons

Given: AC = CD A D

Prove: ABC DEC

A D Given in problem

AC = CD (included side) Given

ACB DCE Vertical Angles Theorem

ASA PostulateABC DEC

Proof: because alternate interior angles are

congruent. By the Midpoint Theorem,

Since vertical angles are congruent,

WRL EDL by ASA.

Write a paragraph proof.

Given: L is the midpoint ofProve: WRL EDL

Write a flow proof.

Given:

Prove:

Proof:

Proof:

Given:

Prove:

Write a flow proof.

STANCES When Ms. Gomez puts her hands on her hips, she forms two triangles with her upper body and arms. Suppose her arm lengths AB and DE measure 9 inches, andAC and EF measure 11 inches. Also suppose that you are given that Determine whether ABC EDF. Justify your answer.

Explore We are given measurements of two sides of each triangle. We need to determine whether the two triangles are congruent.

Plan Since Likewise, We are given Check each possibility using the five methods you know.

Answer: ABC EDF by SSS

Solve We are given information about three sides. Since all three pairs of corresponding sides of the triangles are congruent, ABC EDF by SSS.

Examine You can measure each angle in ABC and EDF to verify that

Answer: ABE CBD by SSS

The curtain decorating the window forms 2 triangles at the top. B is the midpoint of AC. AE = 13 inches and CD = 13 inches. BE and BD each use the same amount of material, 17 inches. Determine whether ABE CBDJustify your answer.

Summary & Homework

• Summary:– If two pairs of corresponding angles and the

included sides of two triangles are congruent, then the triangles are congruent (ASA)

– If two pairs of corresponding angles and a pair of corresponding non-included sides of two triangles are congruent, then the triangles are congruent (AAS)

• Homework: – pg 211 - 212: 15-20 all in two-column proof format