Similar triangles solve problems a tape measure cannot reach. The crew measures the pole outside the loading door from its shadow and a meter stick's shadow, because the two shadow triangles are similar by AA. A line parallel to one side of a triangle divides the other two sides proportionally, and the converse holds. The altitude to the hypotenuse of a right triangle makes two smaller triangles similar to the whole, and that similarity proves the Pythagorean theorem. Congruence and similarity criteria then find lengths and prove relationships in the set's frames.