Question

The four angles of the figure PQRS are right angles. $\angle \mathbf{VTR}$is a right angle. $\mathbf{m}\angle \mathbf{QSR}\mathbf{=}\mathbf{60}$. Find the measures.

a. $\mathbf{m}\angle \mathbf{VTP}$

b.$\mathbf{m}\angle \mathbf{XSY}$

c.$\mathbf{m}\angle \mathbf{RSX}$

d. $\mathbf{m}\angle \mathbf{PSY}$

Short answer

a.The measure of the angle is $\angle VTP$.$\mathbf{90}\mathbf{°}$

b.The measure of the angle is $\angle XSY$.$\mathbf{60}\mathbf{°}$

c.The measure of the angle is $\angle RSX$.$\mathbf{120}\mathbf{°}$

d. The measure of the angle is $\angle PSY$.$\mathbf{90}\mathbf{°}$

Step-by-step solution

Part a. Step 1. Observe the given diagram.

The given diagram is:

Part a. Step 2. Description of step.

It is being given that$\angle VTR$is a right angle.

Therefore, $\angle VTR=90°$.

From the given diagram it can be noticed that the angles $\angle VTR$and $\angle VTP$form the linear pair.

Therefore, the sum of the angles $\angle VTR$and $\angle VTP$is $180°$.

Therefore, it can be obtained that:

$$\begin{gathered} \angle VTR+\angle VTP=180° \\ 90°+\angle VTP=180° \\ \angle VTP=180°-90° \\ \angle VTP=90° \end{gathered}$$

Part a. Step 3. Description of step.

The measure of the angle $\angle VTP$is $90°$.

Part b. Step 1. Observe the given diagram.

The given diagram is:

Part b. Step 2. Description of step.

It is being given that$m\angle QSR=60$.

From the given diagram it can be noticed that the angles$\angle QSR$ and$\angle XSY$ are the vertically opposite angles.

Therefore, $\angle XSY=\angle QSR$.

Therefore, it can be obtained that:

$\begin{array}{c}\angle XSY=\angle QSR\\ =60°\end{array}$

Part b. Step 3. Description of step.

The measure of the angle$\angle XSY$ is$60°$.

Part c. Step 1. Observe the given diagram.

The given diagram is:

Part c. Step 2. Description of step.

From the given diagram it can be noticed that the angles$\angle RSX$ and$\angle XSY$ forms the linear pair.

Therefore, sum of the angles$\angle RSX$ and$\angle XSY$ is$180°$.

Therefore, it can be obtained that:

$$\begin{gathered} \angle RSX+\angle XSY=180° \\ \angle RSX+60°=180° \\ \angle RSX=180°-60° \\ \angle RSX=120° \end{gathered}$$

Part d. Step 3. Description of step.

The measure of the angle$\angle RSX$ is $120°$.

Part d. Step 1. Observe the given diagram.

The given diagram is:

Part b. Step 2. Description of step.

It is being given thatthe four angles of figure$PQRS$ are right angles.

Therefore, it can be said that $\angle PSR=90°$.

From the given diagram it can be noticed that the angles$\angle PSR$ and$\angle PSY$ forms the linear pair.

Therefore, the sum of the angles$\angle PSR$ and$\angle PSY$ is$180°$.

Therefore, it can be obtained that:

$$\begin{gathered} \angle PSR+\angle PSY=180° \\ 90°+\angle PSY=180° \\ \angle PSY=180°-90° \\ \angle PSY=90° \end{gathered}$$

Part d. Step 3. Description of step.

The measure of the angle$\angle PSY$ is $90°$.