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.

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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. 1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

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. 1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Key Concepts, continued 1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

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. 1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Key Concepts, continued 1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

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. 1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

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. 1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

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 1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

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 1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Guided Practice Example 2 Determine whether the triangles are similar. Explain your reasoning. 1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Guided Practice: Example 2, continued Identify the given information. According to the diagram, Given the side lengths, both ∠A and ∠E are included angles. 1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Guided Practice: Example 2, continued Compare the given side lengths of both triangles. If the triangles are similar, then the corresponding sides are proportional. The side lengths are proportional. 1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

✔ Guided Practice: Example 2, continued 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. ✔ 1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Guided Practice: Example 2, continued http://www.walch.com/ei/00157 1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Guided Practice Example 3 Determine whether the triangles are similar. Explain your reasoning. 1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Guided Practice: Example 3, continued Identify the given information. The measures of each side of both triangles are given. 1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Guided Practice: Example 3, continued 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. 1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

✔ Guided Practice: Example 3, continued State your conclusion. Similar triangles must have side lengths that are proportional. is not similar to . ✔ 1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity

Guided Practice: Example 3, continued http://www.walch.com/ei/00158 1.7.1: Proving Triangle Similarity Using SAS and SSS Similarity