Question

In the diagram,$\mathbf{m}\angle \mathbf{VOZ}\mathbf{=}\mathbf{90}$ .

$\overline{\mathrm{OW}}$is an altitude of$△\mathbf{VOZ}$ .

$\overline{\mathrm{OX}}$bisects $\angle \mathbf{VOZ}$.

$\overline{\mathrm{OY}}$is a median of$△\mathbf{VOZ}$ .

Find the measures of the four numbered angles.

$\mathbf{m}\angle \mathbf{Z}\mathbf{=}\mathbf{k}$

Short answer

The measures of the four numbered angles$\angle 1$ ,$\angle 2$,$\angle 3$ and$\angle 4$ are$k°$ ,$45°-k°$,$45°-k°$ and$k°$ respectively.

Step-by-step solution

- Observe the given diagram.

The given diagram is:

- Description of step.

It is being given that$m\angle VOZ=90$.

Therefore it can be obtained that:

$\angle 1+\angle 2+\angle 3+\angle 4=90°$.

As,$\overline{OX}$bisects $\angle VOZ$.

Therefore,$\angle VOZ=2\angle VOX$.

Therefore it can be obtained that:

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

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

As$\overline{OX}$, bisects $\angle VOZ$.

Therefore$\angle VOX=\angle ZOX=45°$,

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

$\angle 1+\angle 2=45°$

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

$\angle 3+\angle 4=45°$

- Description of step.

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

As,$VZ$ is hypotenuse and$\overline{OY}$ 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:

$\angle YOZ=\angle YZO$

Therefore, it can be obtained that:

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

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

- Description of step.

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

$\begin{array}{c}\angle 3+\angle 4=45°\\ \angle 3+k°=45°\\ \angle 3=45°-k°\end{array}$

Therefore, the measure of the angle$\angle 3$ is$45°-k°$ .

- Description of step.

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:

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

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

Therefore,$\angle VWO=90°$ .

- Description of step.

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:

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

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

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

$\begin{array}{c}\angle 1+\angle 2=45°\\ k°+\angle 2=45°\\ \angle 2=45°-k°\end{array}$

Therefore, the measure of the angle$\angle 2$ is$45°-k°$ .