Question

find ( mangle t ).

Explanation:

Step1: Identify the triangle type

This is a right - triangle \( \triangle TUS \) with \( \angle U = 90^{\circ} \), \( TU = 5 \) and \( SU=5\sqrt{3} \). We can use trigonometric ratios to find \( m\angle T \). The tangent of an angle in a right - triangle is defined as the ratio of the opposite side to the adjacent side. For \( \angle T \), the opposite side is \( SU = 5\sqrt{3} \) and the adjacent side is \( TU = 5 \).

Step2: Calculate the tangent of \( \angle T \)

The formula for the tangent of an angle \( \theta \) in a right - triangle is \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \). For \( \angle T \), let \( \theta=\angle T \), then \( \tan(\angle T)=\frac{SU}{TU} \). Substituting the values \( SU = 5\sqrt{3} \) and \( TU = 5 \), we get \( \tan(\angle T)=\frac{5\sqrt{3}}{5}=\sqrt{3} \).

Step3: Find the measure of \( \angle T \)

We know that if \( \tan\theta=\sqrt{3} \), and we are dealing with an acute angle (since it's a triangle angle in a right - triangle, and the other angle is also acute), then from the trigonometric values of special angles, we know that \( \tan(60^{\circ})=\sqrt{3} \). So, \( m\angle T = 60^{\circ} \).

Answer:

\( 60 \)