Assume that AD is the geometric mean of BD and DC in

Two-dimensional figures such as and in Figure 5.3 can be similar, but it is also possible for three-dimensional figures to be similar. Similar orange juice containers are shown in Figures 5.4 a and 5.4 b. Informally, two figures are “similar” if one is an enlargement of the other. Thus a tuna fish can and an orange juice can are not similar, even if both are right-circular cylinders [see Figures 5.4b and 5.4c]. We will consider cylinders in greater detail in Chapter 9. 䉭DEF 䉭ABC a A C B Figure 5.3 b D F E O.J. 6 ounces O.J. 6 ounces Figure 5.4 O.J. 1 6 oun ces O.J. 1 6 ounces T U N A T U N A Our discussion of similarity will generally be limited to plane figures. For two polygons to be similar, it is necessary that each angle of one polygon be congruent to the corresponding angle of the other. However, the congruence of angles is not sufficient to establish the similarity of polygons. The vertices of the congruent an- gles are corresponding vertices of the similar polygons. If in one polygon is con- gruent to in the second polygon, then vertex A corresponds to vertex M, and this is symbolized ; we can indicate that corresponds to by writing . A pair of angles like and are corresponding angles, and the sides determined by consecutive and corresponding vertices are corresponding sides of the similar polygons. For instance, if and , then corresponds to . MN AB B 4 N A 4 M ∠M ∠A ∠A 4 ∠M ∠M ∠A A 4 M ∠M ∠A Discover When a transparency is projected onto a screen, the image created is similar to the projected figure. EXAMPLE 1 Given similar quadrilaterals ABCD and HJKL with congruent angles as indicated in Figure 5.5, name the vertices, angles, and sides that correspond to each other. a B C A D Figure 5.5 b H L J K Solution Because , it follows that Similarly, D 4 L and ∠D 4 ∠L C 4 K and ∠C 4 ∠K B 4 J and ∠B 4 ∠J A 4 H and ∠A 4 ∠H ∠A ⬵ ∠H a b c With an understanding of corresponding angles and corresponding sides, we can define similar polygons. When pairs of consecutive and corresponding vertices are associated, the corresponding sides are included between the corresponding angles or vertices. AB 4 HJ, BC 4 JK, CD 4 KL, and AD 4 HL 쮿 Geometry in Nature The segments of the chambered nautilus are similar not congruent in shape. © Joao VirissimoShutterstock Two polygons are similar if and only if two conditions are satisfied: 1. All pairs of corresponding angles are congruent.

2. All pairs of corresponding sides are proportional. DEFINITION

The second condition for similarity requires that the following extended proportion exists for the sides of the similar quadrilaterals of Example 1. Note that both conditions for similarity are necessary Although condition 1 is satisfied for square EFGH and rectangle RSTU [see Figures 5.6a and b], the figures are not similar—that is, one is not an enlargement of the other—because the extended propor- tion is not true. On the other hand, condition 2 is satisfied for square EFGH and rhom- bus WXYZ [see Figures 5.6a and 5.6c], but the figures are not similar because the pairs of corresponding angles are not congruent. AB HJ = BC JK = CD KL = AD HL G H a F E Figure 5.6 U b R S Z W Y c EXAMPLE 2 Which figures must be similar? a Any two isosceles triangles c Any two rectangles b Any two regular pentagons d Any two squares Solution a No; pairs need not be , nor do the pairs of sides need to be proportional. b Yes; all angles are congruent measure 108° each, and all pairs of sides are proportional. c No; all angles measure 90°, but the pairs of sides are not necessarily proportional. d Yes; all angles measure 90°, and all pairs of sides are proportional. ⬵ ∠ 쮿 Exs. 1–4