IntroductionCongruent 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.

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1.6.1: Defining Similarity

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. 2

1.6.1: Defining Similarity

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.

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1.6.1: Defining Similarity

Key Concepts, continued• The symbol shows that parts are corresponding.

; they are equivalent.

; they are equivalent.

; they are equivalent.

• The corresponding angles are used to name the corresponding sides.

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1.6.1: Defining Similarity

Key Concepts, continued• Observe the diagrams of and .

• The symbol for similarity ( ) is used to show that figures are similar.

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1.6.1: Defining Similarity

Key Concepts, continued

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1.6.1: Defining Similarity

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

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1.6.1: Defining Similarity

Guided Practice

Example 1Use the definition of similarity in terms of similarity transformations to determine whether the two figures are similar. Explain your answer.

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1.6.1: Defining Similarity

Guided Practice: Example 1, continued

1. Examine the orientation of the triangles.

The orientation of the triangles has remained the same, indicating translation, dilation, stretch, or compression.

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1.6.1: Defining Similarity

ABC XYZ

Side Orientation Side Orientation

Top left side of triangle Top left side of triangle

Bottom left side of triangle Bottom left side of triangle

Right side of triangle Right side of triangle

Guided Practice: Example 1, continued

2. 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,

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1.6.1: Defining Similarity

Guided Practice: Example 1, continuedCalculate the distance of .

Distance formula

Substitute (–4, 1) and (1, 4) for (x1, y1) and (x2, y2).

Simplify.

The distance of is units.

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1.6.1: Defining Similarity

Guided Practice: Example 1, continuedCalculate the distance of .

Distance formula

Substitute (1, 4) and (2, –2) for (x1, y1) and (x2, y2).

Simplify.

The distance of is units.

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1.6.1: Defining Similarity

Guided Practice: Example 1, continuedCalculate the distance of .

Distance formula

Substitute (–4, 1) and (2, –2) for (x1, y1) and (x2, y2).

Simplify.

The distance of is units.

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1.6.1: Defining Similarity

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1.6.1: Defining Similarity

Guided Practice: Example 1, continuedCalculate the distance of .

Distance formula

Substitute (–8, 2) and (2, 8) for (x1, y1) and (x2, y2).

Simplify.

The distance of is units.

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1.6.1: Defining Similarity

Guided Practice: Example 1, continuedCalculate the distance of .

Distance formula

Substitute (2, 8) and (4, –4) for (x1, y1) and (x2, y2).

Simplify.

The distance of is units.

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1.6.1: Defining Similarity

Guided Practice: Example 1, continuedCalculate the distance of .

Distance formula

Substitute (–8, 2) and (4, –4) for (x1, y1) and (x2, y2).

Simplify.

The distance of is units.

Guided Practice: Example 1, continued

3. 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.

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1.6.1: Defining Similarity

Guided Practice: Example 1, continued

4. 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. 18

1.6.1: Defining Similarity

Guided Practice: Example 1, continued

5. State your conclusion.A dilation is a similarity transformation; therefore, and are similar. The ratio of similitude is 2.

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1.6.1: Defining Similarity

✔

Guided Practice: Example 1, continued

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1.6.1: Defining Similarity

Guided Practice

Example 2Use the definition of similarity in terms of similarity transformations to determine whether the two figures are similar. Explain your answer.

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1.6.1: Defining Similarity

Guided Practice: Example 2, continued

1. 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 .

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1.6.1: Defining Similarity

Guided Practice: Example 2, continued

2. 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.

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1.6.1: Defining Similarity

✔

Guided Practice: Example 2, continued

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1.6.1: Defining Similarity