Question

the two triangles below are similar. also, ( m angle d = 15^{circ} ) and ( m angle e = 95^{circ} ) as shown below. find ( m angle r, m angle s ), and ( m angle t ). assume the triangles are accurately drawn.

Explanation:

Step1: Find \(m\angle F\)

In \(\triangle DEF\), use the triangle - angle - sum theorem (\(m\angle D+m\angle E + m\angle F=180^{\circ}\)).
Substitute \(m\angle D = 15^{\circ}\) and \(m\angle E=95^{\circ}\) into the formula:
\(m\angle F=180^{\circ}-(m\angle D + m\angle E)=180^{\circ}-(15^{\circ}+95^{\circ})=180^{\circ}-110^{\circ}=70^{\circ}\)

Step2: Use the property of similar triangles

Since the two triangles are similar, corresponding angles are equal.
\(\angle R\) corresponds to \(\angle E\), \(\angle S\) corresponds to \(\angle F\), and \(\angle T\) corresponds to \(\angle D\).
So \(m\angle R = m\angle E = 95^{\circ}\), \(m\angle S=m\angle F = 70^{\circ}\), \(m\angle T=m\angle D=15^{\circ}\)

Answer:

\(m\angle R = 95^{\circ}\), \(m\angle S = 70^{\circ}\), \(m\angle T = 15^{\circ}\)