Question

the two triangles below are similar.
also, ( mangle t = 15^{circ} ) and ( mangle s = 110^{circ} ) as shown below.
find ( mangle d ), ( mangle e ), and ( mangle f ).
assume the triangles are accurately drawn.
( mangle d=square^{circ} )
( mangle e=square^{circ} )
( mangle f=square^{circ} )

Explanation:

Step1: Find \(m\angle R\)

In \(\triangle RST\), use the angle - sum property of a triangle (\(m\angle R+m\angle S + m\angle T=180^{\circ}\)).
Substitute \(m\angle T = 15^{\circ}\) and \(m\angle S=110^{\circ}\):
\(m\angle R=180^{\circ}-(110^{\circ}+15^{\circ})=180^{\circ}-125^{\circ}=55^{\circ}\)

Step2: Use the property of similar triangles

Since \(\triangle RST\sim\triangle DEF\), corresponding angles are equal.
\(m\angle D=m\angle R = 55^{\circ}\), \(m\angle E=m\angle S=110^{\circ}\), \(m\angle F=m\angle T = 15^{\circ}\)

Answer:

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