Question

the two triangles below are similar. also, m∠r=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 \(180^\circ\). First, find the measure of \(\angle S\) in \(\triangle RST\).
The sum of angles in a triangle is \(180^\circ\), so \(m\angle S=180^\circ - m\angle R - m\angle T\).
Substitute \(m\angle R = 15^\circ\) and \(m\angle T = 110^\circ\):
\(m\angle S=180^\circ - 15^\circ - 110^\circ=55^\circ\).

Step2: Determine corresponding angles in similar triangles

Assuming the correspondence of vertices is \(R
ightarrow E\), \(S
ightarrow D\), \(T
ightarrow F\) (or based on the diagram's shape, the angles should correspond as follows: \(\angle T\) corresponds to \(\angle D\), \(\angle R\) corresponds to \(\angle F\), \(\angle S\) corresponds to \(\angle E\))? Wait, actually, looking at the diagram, \(\angle T\) is a larger angle, \(\angle R\) is a small angle, so in the second triangle, \(\angle D\) should correspond to \(\angle T\), \(\angle F\) corresponds to \(\angle R\), and \(\angle E\) corresponds to \(\angle S\).

So:

• \(m\angle D = m\angle T = 110^\circ\) (corresponding angles in similar triangles)
• \(m\angle F = m\angle R = 15^\circ\) (corresponding angles in similar triangles)
• \(m\angle E = m\angle S = 55^\circ\) (corresponding angles in similar triangles, and also since sum of angles in \(\triangle DEF\) is \(180^\circ\), \(180 - 110 - 15 = 55\))

Answer:

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