SECTION A.2 Geometry Essentials A17 Figure 17 Congruent triangles b a c e d f 1005 305 505 305 505 1005 Figure 18 805 10 (a) (b) (c) 805 10 15 7 8 8 20 15 8 20 405 405 405 7 40 8 5 It is not necessary to verify that all three angles and all three sides are the same measure to determine whether two triangles are congruent. Determining Congruent Triangles • Angle–Side–Angle Case Two triangles are congruent if two of the angles are equal and the lengths of the corresponding sides between the two angles are equal. For example, in Figure 18(a), the two triangles are congruent because two angles and the included side are equal. • Side–Side–Side Case Two triangles are congruent if the lengths of the corresponding sides of the triangles are equal. For example, in Figure 18(b), the two triangles are congruent because the three corresponding sides are all equal. • Side–Angle–Side Case Two triangles are congruent if the lengths of two corresponding sides are equal and the angles between the two sides are the same. For example, in Figure 18(c), the two triangles are congruent because two sides and the included angle are equal. We contrast congruent triangles with similar triangles. DEFINITION Similar Triangles Two triangles are similar if the corresponding angles are equal and the lengths of the corresponding sides are proportional. In Words Two triangles are similar if they have the same shape, but (possibly) different sizes. For example, the triangles in Figure 19 (on the next page) are similar because the corresponding angles are equal. In addition, the lengths of the corresponding sides are proportional because each side in the triangle on the right is twice as long as each corresponding side in the triangle on the left. That is, the ratio of the corresponding sides is a constant: d a e b f c 2. = = =
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