Exercise 34 Page 291

Let's look at the given figure.

To prove that $\angle X$ and $\angle YZV$ are complementary, we will construct a two-column proof. Let's start with stating the given information and the statement that we will prove as our first step. The Isosceles Triangle Theorem states that if two sides of a triangle are congruent, then the angles opposite those sides are congruent. Using the theorem, we can write the following step. Looking at the figure, we see that $\angle WVX$ and $\angle YVZ$ are vertical angles. In this case, we will use the Vertical Angles Theorem to write the third step. Next, we will use the Transitive Property between second and third step to write the next step. By the definition of congruent angles, two angles are congruent if and only if they have the same measure of angle. Given that $\overline{ZY}\perp \overline{YV},$ we can write the following step. Using the definition of right a triangle, let's write the seventh step. We know that the acute angles of a right triangle are complementary. By the definition of complementary angles, the sum of the measures of complementary angles is $90^{\circ }.$ From the fifth step, we will substitute $\angle X$ for $\angle YVZ$ into the equation. Finally, using the definition of complementary angles one more time, we can complete the proof. Combining these steps, let's construct the two-column proof.

Statements	Reasons
$△XWV$ is isosceles; $\overline{ZY}\perp \overline{YV}$	Given
$\angle X\cong \angle WVX$	Isosceles Triangle Theorem
$\angle WVX\cong \angle YVZ$	Vertical Angles Theorem
$\angle X\cong \angle YVZ$	Transitive Property
$m\angle X=m\angle YVZ$	Definition of Congruent Angles
$m\angle YVZ=90^{\circ }$	Perpendicular lines form right angle
$△ZVY$ is a right triangle.	Definition of Right Triangle
$\angle YZV$ and $\angle YVZ$ are complementary.	The acute angles of a right triangle are complementary
$m\angle YZV+m\angle YVZ=90^{\circ }$	Definition of Complementary Angles
$m\angle YZV+m\angle X=90^{\circ }$	Substitution Property
$\angle X$ and $\angle YZV$ are complementary.	Definition of Complementary Angles