Exercise 11 Page 457

Let's make a diagram illustrating the situation.

Finding the Height

The triangles have two corresponding angles that are congruent. Then, by the Angle-Angle Similarity Theorem the triangles are similar. For similar polygons corresponding sides are proportional. We can use this information to find the height of the Unisphere. The Unisphere is $140$ feet high.

Why This Works

A light ray that hits a mirror is reflected. The angle between the reflected ray and a line perpendicular to the surface is the same as that between the line and the incoming ray, but on the other side of the line.

The angle measures of ${\theta }_i$ and ${\theta }_r$ are equal and both are complementary to the angles between the ray and the mirror. Therefore, we know that the angles between the ray and the mirror must be congruent.

For the method to work we need to have two triangles that are similar. Two triangles are similar if they have two congruent angles. Let's recall the other pair of congruent angles.

These angles are defined by the plane of the mirror and they are congruent only if the mirror is horizontal. The method works because the mirror lies horizontally.