Similar Triangles: Congruent polygons have the same size and shape, while similar figures have the same shape but may have different sizes. The triangles below are similar. Two polygons are similar if there is a one-to-one correspondence between their vertices such that: 1) All pairs of corresponding angles are congruent. 2) The ratios of the lengths of all pairs of corresponding sides are equal. The symbol to show similarity is ~ *** If you know that two angles of one triangle are congruent to two angles of another triangle, you can conclude that the two triangles are similar. When two triangles are similar, the ratio of the lengths of a pair of corresponding sides is the same as the ratio of any other pair of corresponding linear measurements of the two triangles. Therefore: 1) The lengths of the corresponding sides are in proportion. 2) The ratio of the lengths of any pair of corresponding sides is the same as the ratio of corresponding perimeters, altitudes, or medians. Examples: 1) At a certain time during the day, light falls so that a pole 10 feet in height casts a shadow 15 feet in length on level ground. At the same time a man casts a shadow that is 9 feet in length. How tall is the man? 2) The length of the shortest side of a triangle is 12, and the length of the shortest side of a similar triangle is 5. If the longest side of the larger triangle is 15, what is the longest side of the smaller triangle? a) 2.4 b) 6.25 c) 24 d) 36 3) A person 5 feet tall is standing near a tree 30 feet high. If the length of the person’s shadow is 3 feet, what is the length of the shadow of the tree?