161
Chapter 6: Similar Figures
Example Problems These problems show the answers and solutions. Tell whether the figures in each pair are similar. If so, tell why; if not, tell why not. 1. 12
T
S
8
X
W
8 Q
R
8 U
V
Answer: No The corresponding angles are equal, but only two corresponding sides are in proportion. The vertical sides of both figures are in 1:1 ratio, but the horizontal sides are in 3:2 ratio. All must be in the same ratio for the figures to be similar. 2. 8
A
B
12 D
E 4 H
F G
6
C
Answer: Yes
The corresponding angles are equal; after all, they’re all right angles. The AB = BC = CD = AD = 1 . corresponding sides are all in 2:1 ratio: EH EF FG GH 2
3. S 10
O 6 L
10
20
R
N
6 120° M
12
P
12
20
Q
Answer: No Here the corresponding sides are all in the ratio 1:2. That’s a good sign. However, the first figure, LMNO is a parallelogram with opposite angles of 120° and 60°. PQRS is a rectangle with 90° angles. The corresponding angles are not equal, so the figures are not similar.
Similar Triangles As a general rule, in order to prove that two polygons are similar, it is necessary to prove that all corresponding angles are equal and all corresponding sides are in proportion (that is the ratios between them are all equal). Triangles, however, are a special case. Postulate 18: If two angles of one triangle are equal to two angles of a second triangle, then the triangles are similar. (This is known as the AA Triangle Similarity Postulate.)