Question

Use the diagram to complete each statement..

$\begin{array}{l}Ifm\angle R=30,thenm\angle RSU=?,\\ m\angle TSU=?,andm\angle T=?\end{array}$

Short answer

The measure of angles,

$\begin{array}{l}m\angle RSU={60}^{\circ }\\ m\angle TSU={30}^{\circ }\\ m\angle T={60}^{\circ }\end{array}$.

Step-by-step solution

Step 1. Given information.    

Figure is given.

$m\angle R=30°$

Step 2. Concept used.

Sum of the internal angles in a triangle is ${180}^{\circ }$

Step 3. Now find radii.

In triangle RSU,

$\begin{array}{c}m\angle \text{\hspace{0.17em}}R+m\angle \text{}RSU+m\angle \text{}RUS={180}^{\circ }\\ {30}^{\circ }+m\angle \text{}RSU+{90}^{\circ }={180}^{\circ }\\ m\angle \text{}RSU+{120}^{\circ }={180}^{\circ }\\ m\angle \text{}RSU={60}^{\circ }\end{array}$

Now

$\begin{array}{c}m\angle \text{\hspace{0.17em}}RSU+m\angle \text{\hspace{0.17em}}TSU={90}^{\circ }\\ {60}^{\circ }+m\angle \text{\hspace{0.17em}}TSU={90}^{\circ }\\ m\angle \text{\hspace{0.17em}}TSU={30}^{\circ }\end{array}$

In triangle STU,

$\begin{array}{c}m\angle \text{\hspace{0.17em}}SUT+m\angle \text{\hspace{0.17em}}TSU+m\angle \text{\hspace{0.17em}}T={180}^{\circ }\\ {90}^{\circ }+{30}^{\circ }+m\angle \text{\hspace{0.17em}}T={180}^{\circ }\\ {120}^{\circ }+m\angle \text{\hspace{0.17em}}T={180}^{\circ }\\ m\angle \text{\hspace{0.17em}}T={60}^{\circ }\end{array}$

Therefore, the measure of $m\angle RSU=60°,m\angle TSU=60°,m\angle T=60°$.