Slide 1Copyright © 2015, 2011, 2008 Pearson Education, Inc.

Chapter 2

Ratio, Proportion, and Triangle Applications

Chapter 6

Slide 2Copyright © 2015, 2011, 2008 Pearson Education, Inc.

Congruent and Similar Triangles

Section 6.5

Slide 3Copyright © 2015, 2011, 2008 Pearson Education, Inc.

Two triangles are congruent when they have the same shape and the same size. Corresponding angles are equal, and corresponding sides are equal.

Congruent Triangles

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Angle-Side-Angle (ASA)

If the measures of two angles of a triangle equal the measures of two angles of another triangle, and the lengths of the sides between each pair of angles are equal, the triangles are congruent.

For example, these two triangles are congruent by Angle-Side-Angle.

Slide 5Copyright © 2015, 2011, 2008 Pearson Education, Inc.

Side-Side-Side (SSS)

If the lengths of the three sides of a triangle equal the lengths of the corresponding sides of another triangle, the triangles are congruent.

For example, these two triangles are congruent by Side-Side-Side.

Slide 6Copyright © 2015, 2011, 2008 Pearson Education, Inc.

Side-Angle-Side (SAS)

If the lengths of two sides of a triangle equal the lengths of corresponding sides of another triangle, and the measures of the angles between each pair of sides are equal, the triangles are congruent.

For example, these two triangles are congruent by Side-Angle-Side.

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Example

Determine whether triangle MNO is congruent to triangle RQS.

Since the lengths of all three sides of triangle MNO equal the lengths of all three sides of triangle RQS, the triangles are congruent.

Slide 8Copyright © 2015, 2011, 2008 Pearson Education, Inc.

Example

Determine whether triangle GHI is congruent to triangle JKL.

The triangles are NOT congruent. The angle measures are not the same.

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Similar triangles are found in art, engineering, architecture, biology, and chemistry. Two triangles are similar when they have the same shape but not necessarily the same size.

Similar Triangles

Slide 10Copyright © 2015, 2011, 2008 Pearson Education, Inc.

In similar triangles, the measures of corresponding angles are equal and corresponding sides are in proportion.

a = 3

c = 8

b = 5 d = 6 e = 10

f = 16

Side a corresponds to side d, side b corresponds to side e, and side c corresponds to side f.

Similar Triangles

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Example

Find the ratio of corresponding sides for the similar triangles QRS and XYZ.

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Example

Given that the triangles are similar, find the missing length x.

Since the triangles are similar,corresponding sides are in proportion.

Slide 13Copyright © 2015, 2011, 2008 Pearson Education, Inc.

ExampleTammy Shultz, a firefighter, needs to estimate the height of a burning building. She estimates the length of her shadow to be 8 feet long and the length of the building’s shadow to be 60 feet long. Find the approximate height of the building if she is 5 feet tall.

Slide 14Copyright © 2015, 2011, 2008 Pearson Education, Inc.

Example

The height of the building is about 37.5 feet.

5

8 60

n

5 60 8 n 300 8n

300 8

8 8

n

37.5 n