###### Example 1.16

The two triangles below are congruent. List the corresponding parts, and find the angles \(\theta\text{,}\) \(\phi\text{,}\) and \(\chi\text{,}\) and side \(z\text{.}\)

In these triangles, \(\angle B=\angle D\) because they are both right angles, and \(\angle BCA=\angle DCE\) because they are vertical angles, so \(\theta =25\degree\text{.}\) The third angles, \(\angle A\) and \(\angle E\text{,}\) must also be equal, so \(\phi = \chi =65\degree\text{.}\) (Do you see why?) The sides opposite each pair of corresponding angles are equal, so \(AB=DE\text{,}\) \(BC=CD\) and \(AC=CE\text{.}\) In particular, we find that \(z=9\text{.}\)