91Ó°ÊÓ

The given diagram is:

It is being given that$m∠VOZ=90$.

Therefore it can be obtained that:

$∠1+∠2+∠3+∠4=90°$.

As,$\stackrel{¯}{OX}$bisects $∠VOZ$.

Therefore,$∠VOZ=2∠VOX$.

Therefore it can be obtained that:

$\begin{array}{c}∠VOZ=2∠VOX\\ 90°=2∠VOX\\ \frac{90°}{2}=∠VOX\\ 45°=∠VOX\end{array}$

Therefore, the measure of the angle$∠VOX$is$45°$$45°$.

As$\stackrel{¯}{OX}$, bisects $∠VOZ$.

Therefore$∠VOX=∠ZOX=45°$,

As,$∠VOX=45°$ , therefore it can be obtained that:

$∠1+∠2=45°$

As, $∠ZOX=45°$, therefore it can be obtained that:

$∠3+∠4=45°$

The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.

As,$VZ$ is hypotenuse and$\stackrel{¯}{OY}$ is a median of$△VOZ$ .

Therefore,Y is the midpoint of the hypotenuse and therefore it is equidistant from the three vertices of the triangle.

That implies,$OY=YZ$ .

The angles opposite to the equal sides are also equal.

Therefore, it can be noticed that:

$∠YOZ=∠YZO$

Therefore, it can be obtained that:

$\begin{array}{c}∠YOZ=∠YZO\\ ∠4=∠Z\\ ∠4=k°\end{array}$

Therefore, the measure of the angle $∠4$is$k°$ .

As,$3+4=45°$, therefore it can be obtained that:

$\begin{array}{c}∠3+∠4=45°\\ ∠3+k°=45°\\ ∠3=45°−k°\end{array}$

Therefore, the measure of the angle$∠3$ is$45°−k°$ .

As, the sum of all the angles of a triangle is$180°$ .

Therefore, the sum of the angles of triangleVOZ is $180°$.

Therefore, it can be obtained that:

$\begin{array}{c}∠VOZ+∠VZO+∠ZVO=180°\\ 90°+k°+∠ZVO=180°\\ ∠ZVO=180°−90°−k°\\ ∠ZVO=90°−k°\end{array}$

As,$\stackrel{¯}{OW}$ is an altitude of $△VOZ$.

Therefore,$∠VWO=90°$ .

As, the sum of all the angles of a triangle is$180°$ .

Therefore, the sum of the angles of triangleVOZ is$180°$ .

Therefore, it can be obtained that:

$\begin{array}{c}∠VOW+∠VWO+∠WVO=180°\\ ∠1+90°+∠ZVO=180°\\ ∠1+90°+90°−k=180°\\ ∠1+180°−k°=180°\\ ∠1=k°\end{array}$

Therefore, the measure of the angle$∠1$ is$k°$ .

As,$∠1+∠2=45°$ , therefore it can be obtained that:

$\begin{array}{c}∠1+∠2=45°\\ k°+∠2=45°\\ ∠2=45°−k°\end{array}$

Therefore, the measure of the angle$∠2$ is$45°−k°$ .