Exercise 14 Page 612

From the diagram, we see an isosceles triangle inside of the right triangle. According to the Base Angles Theorem, if two sides of a triangle are congruent, then the angles opposite them are congruent. Therefore, the second base angle of our isosceles triangle is also $x^{\circ }.$ Let's also label the vertex angle as $z^{\circ }$ in our diagram.

Now we see that $40^{\circ }$ and $z^{\circ }$ form a linear pair. By the Linear Pair Postulate, we know that these angles are supplementary. Therefore, we can write the equation. Let's solve this equation. Knowing the measure of the vertex angle, we can by using the Triangle Sum Theorem, write the following equation. Let's solve the equation. To find $y^{\circ },$ we add the measure of $x^{\circ }$ to our diagram and highlight relevant parts. Again, we can use the Triangle Sum Theorem to write an equation that includes the angle of the right triangle. Let's solve the equation.