Question

prove: ∠xvz≅∠wvy. we are given an image of wz and xy which intersected at point v. m∠xvz + m∠zvy = 180° by the definition of supplementary angles. since the sum of m∠zvy + m∠wvy = 180° by the definition of supplementary angles, m∠zvy can be subtracted from both sides of the equation because of the subtraction property of equality. therefore, m∠xvz = m∠wvy and ∠xvz≅∠wvy by the definition of congruent angles. alternate interior angles theorem, corresponding angles theorem, vertical angles theorem, same - side interior angles theorem

Explanation:

Step1: Recall supplementary - angle property

Given that $\angle XVZ+\angle ZVY = 180^{\circ}$ and $\angle ZVY+\angle WVY=180^{\circ}$ because $\overline{WZ}$ and $\overline{XY}$ intersect at point $V$.

Step2: Use the subtraction property of equality

From $\angle XVZ+\angle ZVY = 180^{\circ}$ and $\angle ZVY+\angle WVY = 180^{\circ}$, we can subtract $\angle ZVY$ from both equations.
Let $a=\angle XVZ+\angle ZVY$, $b = 180^{\circ}$, $c=\angle ZVY+\angle WVY$. Since $a = b$ and $c = b$, then $a - \angle ZVY=c-\angle ZVY$.
So, $\angle XVZ=\angle WVY$.

Step3: Use the definition of congruent angles

Since $\angle XVZ=\angle WVY$, by the definition of congruent angles (angles with equal measures are congruent), we have $\angle XVZ\cong\angle WVY$.

Answer:

$\angle XVZ\cong\angle WVY$ is proved by using the definition of supplementary - angles and the subtraction property of equality.