Quadrilaterals ABCD and EFGH are congruent (ABCD ≅ EFGH). Find the measure of each missing angle and side.

The shapes have been rotated, so we need to read the order of the points to find the corresponding angles (ABCD ≅ EFGH). This means that A corresponds to E, B corresponds to F, C corresponds to G, and D corresponds to H. Since the corresponding angles are congruent:

This is also true for the corresponding sides:

Triangles ABC and DEF are similar (ABC ~ DEF). What is the length of side DF?

Here the shapes have not been rotated. A corresponds to D, B corresponds to E, and so on.

When the shapes are similar, the ratio of the sides are proportional. Be careful

The sun casts a shadow on two trees in a field. The smaller of the two is 10.5 ft high and has a shadow 11.25 ft long. The shadow of the taller tree is 17.5 ft long. How tall is that tree?

This is actually how the heights of many tall structures are estimated!

When the sun rays hit the trees and cast a shadow it creates proportional triangles. To solve this problem, all we need to do is solve for the missing height.

So that tree is a whopping 16 ⅓ feet tall.

Pentagons OPQRS and TUVWX are both regular. Find the length of the apothem of TUVWX.

The apothem is the distance from the center of a regular polygon to the midpoint of one side.

Two regular pentagons are always similar, since all the angles are congruent. Not only are their sides proportional, but their heights and apothems will be proportional too.