Question

the two triangles below are similar. also, m∠s = 15° and m∠t = 110° as shown below. find m∠d, m∠e, and m∠f. assume the triangles are accurately drawn.

Explanation:

Step1: Recall properties of similar triangles

Similar triangles have corresponding angles equal. Also, the sum of angles in a triangle is \(180^\circ\).

Step2: Find \(m\angle R\) in \(\triangle RST\)

In \(\triangle RST\), we know \(m\angle S = 15^\circ\) and \(m\angle T=110^\circ\). Using the angle - sum property of a triangle (\(m\angle R + m\angle S+m\angle T = 180^\circ\)), we can find \(m\angle R\) as follows:
\(m\angle R=180^\circ-(m\angle S + m\angle T)\)
\(m\angle R = 180^\circ-(15^\circ + 110^\circ)=180^\circ - 125^\circ = 55^\circ\)

Step3: Determine corresponding angles in similar triangles

Since \(\triangle RST\sim\triangle FDE\) (assuming the order of the letters represents the correspondence of vertices), the corresponding angles are equal.

• \(\angle T\) corresponds to \(\angle D\), so \(m\angle D=m\angle T = 110^\circ\)
• \(\angle R\) corresponds to \(\angle E\), so \(m\angle E=m\angle R = 55^\circ\)
• \(\angle S\) corresponds to \(\angle F\), so \(m\angle F=m\angle S = 15^\circ\)

Answer:

\(m\angle D=\boldsymbol{110^\circ}\), \(m\angle E=\boldsymbol{55^\circ}\), \(m\angle F=\boldsymbol{15^\circ}\)