Question

1. in the diagram shown below, lines $overline{zw}$, $overline{ux}$ and ray $overline{yv}$ all intersect at point y. angle $angle wyv$ is a right angle. if $mangle uyv = 54^{circ}$, determine the measure of each of the following angles. $mangle vyz$ $mangle zyu$ $mangle xyz$ $mangle xyw$ $mangle uyw$

Explanation:

Step1: Find $m\angle VYZ$

Since $\angle WYZ$ is a straight - angle ($180^{\circ}$) and $\angle WYV = 90^{\circ}$, then $m\angle VYZ=180^{\circ}- 90^{\circ}=90^{\circ}$.

Step2: Find $m\angle ZYU$

We know that $m\angle UYV = 54^{\circ}$ and $m\angle VYZ = 90^{\circ}$, so $m\angle ZYU=m\angle VYZ - m\angle UYV=90^{\circ}-54^{\circ}=36^{\circ}$.

Step3: Find $m\angle XYZ$

$\angle XYZ$ and $\angle UYV$ are vertical angles. Vertical angles are equal. So $m\angle XYZ = m\angle UYV=54^{\circ}$.

Step4: Find $m\angle XYW$

$m\angle XYW=m\angle XYZ + m\angle ZYW$. Since $\angle ZYW = 90^{\circ}$ and $m\angle XYZ = 54^{\circ}$, then $m\angle XYW=54^{\circ}+90^{\circ}=144^{\circ}$.

Step5: Find $m\angle UYW$

$m\angle UYW=m\angle UYV + m\angle VYW$. Since $m\angle UYV = 54^{\circ}$ and $m\angle VYW = 90^{\circ}$, then $m\angle UYW=54^{\circ}+90^{\circ}=144^{\circ}$.

Answer:

$m\angle VYZ = 90^{\circ}$, $m\angle ZYU=36^{\circ}$, $m\angle XYZ = 54^{\circ}$, $m\angle XYW = 144^{\circ}$, $m\angle UYW = 144^{\circ}$