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Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle (AA) Similarity Statement is one of them. The Side-Angle-Side (SAS) and Side-Side-Side (SSS) similarity statements are two more ways to show that triangles are similar. In this lesson, we will prove that triangles are similar using the similarity statements. 1 1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

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Page 1: Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle

IntroductionThere are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle (AA) Similarity Statement is one of them. The Side-Angle-Side (SAS) and Side-Side-Side (SSS) similarity statements are two more ways to show that triangles are similar. In this lesson, we will prove that triangles are similar using the similarity statements.

1

1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Page 2: Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle

Key Concepts• The Side-Angle-Side (SAS) Similarity Statement

asserts that if the measures of two sides of a triangle are proportional to the measures of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar.

• Similarity statements identify corresponding parts just like congruence statements do.

2

1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Page 3: Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle

Key Concepts, continued

3

1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Page 4: Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle

Key Concepts, continued• The Side-Side-Side (SSS) Similarity Statement

asserts that if the measures of the corresponding sides of two triangles are proportional, then the triangles are similar.

4

1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Page 5: Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle

Key Concepts, continued

5

1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Page 6: Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle

Key Concepts, continued• It is important to note that while both similarity and

congruence statements include an SSS and an SAS statement, the statements do not mean the same thing.

• Similar triangles have corresponding sides that are proportional, whereas congruent triangles have corresponding sides that are of the same length.

• Like with the Angle-Angle Similarity Statement, both the Side-Angle-Side and the Side-Side-Side similarity statements can be used to solve various problems.

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1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Page 7: Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle

Key Concepts, continued• The ability to prove that triangles are similar is

essential to solving many problems.

• A proof is a set of justified statements organized to form a convincing argument that a given statement is true.

• Definitions, algebraic properties, and previously proven statements can be used to prove a given statement.

7

1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Page 8: Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle

Key Concepts, continued• There are several types of proofs, such as paragraph

proofs, two-column proofs, and flow diagrams.

• Every good proof includes the following:

• a statement of what is to be proven

• a list of the given information

• if possible, a diagram including the given information

• step-by-step statements that support your reasoning

8

1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Page 9: Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle

Common Errors/Misconceptions• misidentifying congruent parts because of the

orientation of the triangles• misreading similarity statements as congruency

statements• incorrectly creating proportions between corresponding

sides

9

1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Page 10: Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle

Guided Practice

Example 2Determine whether the triangles are similar. Explain your reasoning.

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1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Page 11: Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle

Guided Practice: Example 2, continued

1. Identify the given information. According to the diagram,

Given the side lengths, both ∠A and ∠E are included angles.

11

1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Page 12: Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle

Guided Practice: Example 2, continued

2. Compare the given side lengths of both triangles. If the triangles are similar, then the corresponding sides are proportional.

The side lengths are proportional.

12

1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Page 13: Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle

Guided Practice: Example 2, continued

3. State your conclusion. The measures of two sides of are proportional to the measures of two corresponding sides of , and the included angles are congruent.

by the Side-Angle-Side (SAS) Similarity Statement.

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1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

✔

Page 14: Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle

Guided Practice: Example 2, continued

14

1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

http://www.walch.com/ei/00157
Page 15: Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle

Guided Practice

Example 3Determine whether the triangles are similar. Explain your reasoning.

15

1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Page 16: Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle

Guided Practice: Example 3, continued

1. Identify the given information. The measures of each side of both triangles are given.

16

1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Page 17: Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle

Guided Practice: Example 3, continued

2. Compare the side lengths of both triangles. Pair the lengths of the sides of with the corresponding lengths of the sides of to determine if there is a common ratio.

Notice there is not a common ratio; therefore, the side lengths are not proportional.

17

1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Page 18: Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle

Guided Practice: Example 3, continued

3. State your conclusion. Similar triangles must have side lengths that are proportional.

is not similar to .

18

1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

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Page 19: Introduction There are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The Angle-Angle

Guided Practice: Example 3, continued

19

1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

http://www.walch.com/ei/00158

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