Five-Minute Check (over Lesson 7–3) Mathematical Practices Then/Now Theorem 7.3: SSS Triangle Similarity Theorem 7.4: SAS Triangle Similarity Example 1: Use the SSS Theorem Example 2: Analyze Triangles on the Coordinate Plane Key Concept: Corresponding Parts of Similar Triangles Example 3: Use the SAS Similarity Theorem Example 4: Analyze Triangles on the Coordinate Plane Example 5: Indirect Measurement Lesson Menu

Determine whether the triangles are similar. Justify your answer. A. yes, SSS Similarity B. yes, ASA Similarity C. yes, AA Similarity D. No, sides are not proportional. 5-Minute Check 1

Determine whether the triangles are similar. Justify your answer. A. yes, AA Similarity B. yes, SSS Similarity C. yes, SAS Similarity D. No, sides are not proportional. 5-Minute Check 2

Determine whether the triangles are similar. Justify your answer. A. yes, AA Similarity B. yes, SSS Similarity C. yes, SAS Similarity D. No, angles are not equal. 5-Minute Check 3

Find the width of the river in the diagram. B. 28 m C. 24 m D. 22.4 m 5-Minute Check 4

Mathematical Processes 1 Make sense of problems and persevere in solving them. 2 Reason abstractly and quantitatively. 3 Construct viable arguments and critique the reasoning of others. 4 Model with mathematics. 7 Look for and make use of structure. MP

G.SRT.4 Prove theorems about similarity. Content Standards G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G.SRT.4 Prove theorems about similarity. G.SRT.5 Use congruence and similarity for triangles to solve problems and to prove relationships in geometric figures. MP

You used the AA Similarity Postulate to prove triangles are similar. Use the SSS similarity criterion to prove triangles are similar. Use the SAS similarity criterion to prove triangles are similar. Then/Now

Theorem

Theorem

Use the SSS Theorem Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. A. Example 1A

Answer: ABC ∼ DEC by the SSS Similarity Theorem. Use the SSS Theorem Write a proportion that relates to the corresponding sides. Substitute the lengths of the sides. Simplify   SSS Similarity Theorem Answer: ABC ∼ DEC by the SSS Similarity Theorem. Example 1A

Use the SSS Theorem Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. B. Example 1B

Answer: The sides are not proportional, so MNO WXY. Use the SSS Theorem Write a proportion that relates to the corresponding sides. Substitute the lengths of the sides. Simplify ∆MNO is not similar ∆WXY. SSS Similarity Theorem does not apply. Answer: The sides are not proportional, so MNO WXY. Example 1B

Analyze Triangles on the Coordinate Plane Determine whether DEF with vertices D(–4, 0), E(6, –1), and F(3, 3) is similar to LMN with vertices L(–3, –5), M(15, –7), and N(11, –1). Explain your reasoning. Graph the points and determine the corresponding sides. Write a proportion that relates to the corresponding sides. Find the side lengths by using the distance formula. Example 2

Answer: The sides are not proportional, so DEF LMN. Analyze Triangles on the Coordinate Plane Substitute the lengths of the sides. ∆DEF is not similar ∆LMN. The sides are not proportional. Answer: The sides are not proportional, so DEF LMN. Example 2

Key Concept

Use the SAS Similarity Theorem Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. A. Example 3

Use the SAS Similarity Theorem Do you have enough information to write a proportion that relates to the corresponding sides? No Do the triangles have congruent corresponding angles? No The triangles are not similar because the sides are not proportional and the corresponding angles are not congruent. Answer: The triangles are similar, because the sides are not proportional and the angles are not congruent. Example 3

Use the SAS Similarity Theorem Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. B. Example 3

Answer: MNP ∼ MRS by the SAS Similarity Theorem. Use the SAS Similarity Theorem Write a proportion that relates to the corresponding sides. Substitute the lengths of the sides. ∠M ≅ ∠M The Reflexive Property   SAS Similarity Theorem Answer: MNP ∼ MRS by the SAS Similarity Theorem. Example 3

Analyze Triangles on the Coordinate Plane Determine whether WUV with vertices W(–2, 5), U(2, –6), and V(3, 5) is similar to WXY with vertices X(10, –28), and Y(13, 5). Explain your reasoning. Graph the points and determine the corresponding sides. Write a proportion that relates to the corresponding sides. Find the side lengths by using the distance formula. Example 4

Answer: WUV  WXY by SAS Similarity Analyze Triangles on the Coordinate Plane Substitute the lengths of the sides. Simplify ∠W ≅ ∠W The Reflexive Property SAS Similarity Theorem   Answer: WUV  WXY by SAS Similarity Example 4

Joaquin wants to measure the height of a building Indirect Measurement Joaquin wants to measure the height of a building obstructing his view of the lake. He stands so that the tip of his shadow coincides with the tip of the building’s shadow. He is 5 feet 6 inches tall, and his shadow is 18 feet long at the same time that the building’s shadow is 360 feet long. About how tall is the building? Example 5

Indirect Measurement Write a proportion that relates to the corresponding sides. Solve for x. Answer: 110 feet Example 5