Question

the two triangles below are similar. also, ( m angle c = 30 ^ { circ } ) and ( m angle d = 85 ^ { circ } ) as shown below. find ( m angle w ), ( m angle x ), and ( m angle y ). assume the triangles are accurately drawn. ( m angle w = square ^ { circ } ) ( m angle x = square ^ { circ } ) ( m angle y = square ^ { circ } )

Explanation:

Step1: Use the property of similar triangles

Similar triangles have congruent corresponding angles. So, \(\angle D\) corresponds to \(\angle W\), \(\angle C\) corresponds to \(\angle X\), and \(\angle E\) corresponds to \(\angle Y\).

Step2: Find \(m\angle W\)

Since \(\angle D\) and \(\angle W\) are corresponding angles of similar triangles, \(m\angle W=m\angle D = 85^{\circ}\).

Step3: Find \(m\angle X\)

Since \(\angle C\) and \(\angle X\) are corresponding angles of similar triangles, \(m\angle X=m\angle C=30^{\circ}\).

Step4: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). For \(\triangle DEF\), \(m\angle E=180^{\circ}-m\angle D - m\angle C\). Substitute \(m\angle D = 85^{\circ}\) and \(m\angle C = 30^{\circ}\), we get \(m\angle E=180^{\circ}-85^{\circ}-30^{\circ}=65^{\circ}\).

Step5: Find \(m\angle Y\)

Since \(\angle E\) and \(\angle Y\) are corresponding angles of similar triangles, \(m\angle Y=m\angle E = 65^{\circ}\).

Answer:

\(m\angle W = 85^{\circ}\), \(m\angle X = 30^{\circ}\), \(m\angle Y = 65^{\circ}\)