CHAPTER 8 Geometry Slide 2Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 8.1Basic Geometric Figures 8.2Perimeter 8.3Area 8.4Circles 8.5Volume and Surface Area 8.6Relationships Between Angle Measures 8.7Congruent Triangles and Properties of Parallelograms 8.8Similar Triangles

OBJECTIVES 8.8 Similar Triangles Slide 3Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. aIdentify the corresponding parts of similar triangles and determine which sides of a given pair of triangles have lengths that are proportional. bFind lengths of sides of similar triangles using proportions.

8.8 Similar Triangles a Identify the corresponding parts of similar triangles and determine which sides of a given pair of triangles have lengths that are proportional. Slide 4Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. We know that congruent figures have the same shape and size. Similar figures have the same shape, but are not necessarily the same size.

EXAMPLE 8.8 Similar Triangles a Identify the corresponding parts of similar triangles and determine which sides of a given pair of triangles have lengths that are proportional. 1 Slide 5Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Which pairs of triangles appear to be similar? Pairs (a), (c), and (d) appear to be similar.

8.8 Similar Triangles a Identify the corresponding parts of similar triangles and determine which sides of a given pair of triangles have lengths that are proportional. Slide 6Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Similar triangles have corresponding sides and angles.

EXAMPLE 8.8 Similar Triangles a Identify the corresponding parts of similar triangles and determine which sides of a given pair of triangles have lengths that are proportional. 2 Slide 7Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

8.8 Similar Triangles SIMILAR TRIANGLES Slide 8Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Two triangles are similar if and only if their vertices can be matched so that the corresponding angles are congruent and the lengths of corresponding sides are proportional.

EXAMPLE 8.8 Similar Triangles a Identify the corresponding parts of similar triangles and determine which sides of a given pair of triangles have lengths that are proportional. 4 Slide 9Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. These triangles are similar. Which sides are proportional? It appears that if we match X with U, Y with W, and Z with V, the corresponding angles will be congruent. Thus,

EXAMPLE 8.8 Similar Triangles b Find lengths of sides of similar triangles using proportions. 6 Slide 10Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.8 Similar Triangles b Find lengths of sides of similar triangles using proportions. 6 Slide 11Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Recall that if a transversal intersects two parallel lines, then the alternate interior angles are congruent (Section 6.6). Thus, because they are pairs of alternate interior angles. Since are vertical angles, they are congruent. Thus by definition

EXAMPLE 8.8 Similar Triangles b Find lengths of sides of similar triangles using proportions. 6 Slide 12Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. and the lengths of the corresponding sides are proportional. Thus,

EXAMPLE 8.8 Similar Triangles b Find lengths of sides of similar triangles using proportions. 6 Slide 13Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.8 Similar Triangles b Find lengths of sides of similar triangles using proportions. 7 Slide 14Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. How high is a flagpole that casts a 56- ft shadow at the same time that a 6-ft man casts a 5-ft shadow?

EXAMPLE 8.8 Similar Triangles b Find lengths of sides of similar triangles using proportions. 7 Slide 15Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. If we use the sun’s rays to represent the third side of the triangle in our drawing of the situation, we see that we have similar triangles. Let p = the height of the flagpole. The ratio of 6 to p is the same as the ratio of 5 to 56. Thus we have the proportion

EXAMPLE 8.8 Similar Triangles b Find lengths of sides of similar triangles using proportions. 7 Slide 16Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.