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Introduction Congruent triangles have corresponding parts with angle measures that are the same and side lengths that are the same. If two triangles are congruent, they are also similar. Similar triangles have the same shape, but may be different in size. It is possible for two triangles to be similar but not congruent. Just like with determining congruency, it is possible to determine similarity based on the angle measures and lengths of the sides of the triangles. 1.6.1: Defining Similarity

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Key Concepts To determine whether two triangles are similar, observe the angle measures and the side lengths of the triangles. When a triangle is transformed by a similarity transformation (a rigid motion [reflection, translation, or rotation] followed by a dilation), the result is a triangle with a different position and size, but the same shape. If two triangles are similar, then their corresponding angles are congruent and the measures of their corresponding sides are proportional, or have a constant ratio. 1.6.1: Defining Similarity

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**Key Concepts, continued**

The ratio of corresponding sides is known as the ratio of similitude. The scale factor of the dilation is equal to the ratio of similitude. Similar triangles with a scale factor of 1 are congruent triangles. Like with congruent triangles, corresponding angles and sides can be determined by the order of the letters. If is similar to , the vertices of the two triangles correspond in the same order as they are named. 1.6.1: Defining Similarity

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**Key Concepts, continued**

The symbol shows that parts are corresponding. ; they are equivalent. The corresponding angles are used to name the corresponding sides. 1.6.1: Defining Similarity

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**Key Concepts, continued**

Observe the diagrams of and The symbol for similarity ( ) is used to show that figures are similar. 1.6.1: Defining Similarity

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**Key Concepts, continued**

1.6.1: Defining Similarity

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**Common Errors/Misconceptions**

incorrectly identifying corresponding parts of triangles assuming corresponding parts indicate congruent parts assuming alphabetical order indicates congruence changing the order of named triangles, causing parts to be incorrectly interpreted as congruent 1.6.1: Defining Similarity

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**Guided Practice Example 1**

Use the definition of similarity in terms of similarity transformations to determine whether the two figures are similar. Explain your answer. 1.6.1: Defining Similarity

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**Guided Practice: Example 1, continued **

Examine the orientation of the triangles. The orientation of the triangles has remained the same, indicating translation, dilation, stretch, or compression. ABC XYZ Side Orientation Top left side of triangle Bottom left side of triangle Right side of triangle 1.6.1: Defining Similarity

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**Guided Practice: Example 1, continued **

Determine whether a dilation has taken place by calculating the scale factor. First, identify the vertices of each triangle. A (–4, 1), B (1, 4), and C (2, –2) X (–8, 2), Y (2, 8), and Z (4, –4) Then, find the length of each side of and using the distance formula, 1.6.1: Defining Similarity

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**Guided Practice: Example 1, continued**

Calculate the distance of Distance formula Substitute (–4, 1) and (1, 4) for (x1, y1) and (x2, y2). Simplify. The distance of is units. 1.6.1: Defining Similarity

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**Guided Practice: Example 1, continued**

Calculate the distance of Distance formula Substitute (1, 4) and (2, –2) for (x1, y1) and (x2, y2). Simplify. The distance of is units. 1.6.1: Defining Similarity

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**Guided Practice: Example 1, continued**

Calculate the distance of Distance formula Substitute (–4, 1) and (2, – 2) for (x1, y1) and (x2, y2). Simplify. The distance of is units. 1.6.1: Defining Similarity

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**Guided Practice: Example 1, continued**

Calculate the distance of Distance formula Substitute (–8, 2) and (2, 8) for (x1, y1) and (x2, y2). Simplify. The distance of is units. 1.6.1: Defining Similarity

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**Guided Practice: Example 1, continued**

Calculate the distance of Distance formula Substitute (2, 8) and (4, –4) for (x1, y1) and (x2, y2). Simplify. The distance of is units. 1.6.1: Defining Similarity

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**Guided Practice: Example 1, continued**

Calculate the distance of Distance formula Substitute (–8, 2) and (4, – 4) for (x1, y1) and (x2, y2). Simplify. The distance of is units. 1.6.1: Defining Similarity

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**Guided Practice: Example 1, continued **

Calculate the scale factor of the changes in the side lengths. Divide the side lengths of by the side lengths of The scale factor is constant between each pair of corresponding sides. 1.6.1: Defining Similarity

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**Guided Practice: Example 1, continued **

Determine if another transformation has taken place. Multiply the coordinate of each vertex of the preimage by the scale factor, k. Dk(x, y) = (kx, ky) Dilation by a scale factor of k D2(–4, 1) = [2(–4), 2(1)] = (–8, 2) D2(1, 4) = [2(1), 2(4)] = (2, 8) D2(2, –2) = [2(2), 2(–2)] = (4, –4) You can map onto by the dilation with a scale factor of 2. 1.6.1: Defining Similarity

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**✔ Guided Practice: Example 1, continued State your conclusion.**

A dilation is a similarity transformation; therefore, and are similar. The ratio of similitude is 2. ✔ 1.6.1: Defining Similarity

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**Guided Practice: Example 1, continued**

1.6.1: Defining Similarity

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**Guided Practice Example 2**

Use the definition of similarity in terms of similarity transformations to determine whether the two figures are similar. Explain your answer. 1.6.1: Defining Similarity

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**Guided Practice: Example 2, continued **

Examine the angle measures of the triangles. Use a protractor or construction methods to determine if corresponding angles are congruent. None of the angles of are congruent to the angles of 1.6.1: Defining Similarity

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**✔ Guided Practice: Example 2, continued**

Summarize your findings. Similarity transformations preserve angle measure. The angles of and are not congruent. There are no sequences of transformations that will map onto and are not similar triangles. ✔ 1.6.1: Defining Similarity

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**Guided Practice: Example 2, continued**

1.6.1: Defining Similarity

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