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.

$\mathbf{m}\angle \mathbf{Z}\mathbf{=}\mathbf{30}$

Short answer

The measures of the four numbered angles $\angle 1$,$\angle 2$ ,$\angle 3$ and $\angle 4$ are$30°$ ,$15°$ , 15 and 30 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 $45°$.

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, 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:

$\begin{array}{c}\angle YOZ=\angle YZO\\ \angle 4=\angle Z\\ \angle 4=30°\end{array}$

Therefore, the measure of the angle $\angle 4$ is $30°$.

- Description of step.

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

$\begin{array}{c}\angle 3+\angle 4=45°\\ \angle 3+30°=45°\\ \angle 3=45°-30°\\ \angle 3=15°\end{array}$

Therefore, the measure of the angle $\angle 3$ is $15°$.

- Description of step.

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

Therefore, the sum of the angles of triangle$VOZ$ is $180°$.

Therefore, it can be obtained that:

$\begin{array}{c}\angle VOZ+\angle VZO+\angle ZVO=180°\\ 90°+30°+\angle ZVO=180°\\ \angle ZVO+120°=180°\\ \angle ZVO=180°-120°\\ \angle ZVO=60°\end{array}$

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 triangle $VWO$ is$180°$ .

Therefore, it can be obtained that:

$\begin{array}{c}\angle VOW+\angle VWO+\angle WVO=180°\\ \angle 1+90°+\angle ZVO=180°\\ \angle 1+90°+60°=180°\\ \angle 1+150°=180°\\ \angle 1=180°-150°\\ \angle 1=30°\end{array}$

Therefore, the measure of the angle $\angle 1$ is $30°$.

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

$\begin{array}{c}\angle 1+\angle 2=45°\\ 30°+\angle 2=45°\\ \angle 2=45°-30°\\ \angle 2=15°\end{array}$

Therefore, the measure of the angle $\angle 2$ is $15°$.