Question

the two triangles below are similar.
also, ( mangle d = 15^circ ) and ( mangle e = 105^circ ) as shown below.
find ( mangle r ), ( mangle s ), and ( mangle t ).
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 third angle in triangle \(DEF\). Let \(\angle F\) be the third angle. So, \(m\angle D + m\angle E + m\angle F = 180^\circ\). Substituting the known values: \(15^\circ+ 105^\circ + m\angle F=180^\circ\).

Step2: Calculate \(m\angle F\)

Simplify the left - hand side: \(120^\circ+m\angle F = 180^\circ\). Then, \(m\angle F=180^\circ - 120^\circ=60^\circ\).

Step3: Determine corresponding angles in similar triangles

Assuming the correspondence of the triangles is \(\triangle DEF\sim\triangle RTS\) (by looking at the shape and angle positions, \(\angle D\) corresponds to \(\angle S\), \(\angle E\) corresponds to \(\angle R\), and \(\angle F\) corresponds to \(\angle T\)).

• Since \(\angle E\) and \(\angle R\) are corresponding angles, \(m\angle R=m\angle E = 105^\circ\).
• Since \(\angle D\) and \(\angle S\) are corresponding angles, \(m\angle S=m\angle D = 15^\circ\).
• Since \(\angle F\) and \(\angle T\) are corresponding angles, \(m\angle T=m\angle F = 60^\circ\).

Answer:

\(m\angle R = 105^\circ\), \(m\angle S = 15^\circ\), \(m\angle T = 60^\circ\)