Question

the two triangles below are similar. also, ( m angle y = 60^{circ} ) and ( m angle w = 90^{circ} ) as shown below. find ( m angle c, m angle d ), and ( m angle e ). assume the triangles are accurately drawn. ( \begin{array} { l } { m angle c = square ^ { circ } } \\ { m angle d = square ^ { circ } } \\ { m angle e = square ^ { circ } } end{array} )

Explanation:

Step1: Use the property of similar triangles

Similar triangles have equal corresponding angles.

Step2: Identify corresponding angles

\(\angle Y\) corresponds to \(\angle E\), \(\angle W\) corresponds to \(\angle D\), \(\angle X\) corresponds to \(\angle C\).

Step3: Find \(m\angle D\)

Since \(\angle W = 90^{\circ}\) and \(\angle W\) corresponds to \(\angle D\), so \(m\angle D=90^{\circ}\).

Step4: Find \(m\angle E\)

Since \(\angle Y = 60^{\circ}\) and \(\angle Y\) corresponds to \(\angle E\), so \(m\angle E = 60^{\circ}\).

Step5: Use the angle - sum property of a triangle

The sum of angles in a triangle is \(180^{\circ}\). For \(\triangle CED\), \(m\angle C+m\angle D+m\angle E=180^{\circ}\). Substitute \(m\angle D = 90^{\circ}\) and \(m\angle E=60^{\circ}\) into the equation: \(m\angle C+90^{\circ}+60^{\circ}=180^{\circ}\), then \(m\angle C=180^{\circ}-(90^{\circ} + 60^{\circ})=30^{\circ}\).

Answer:

\(m\angle C = 30^{\circ}\), \(m\angle D = 90^{\circ}\), \(m\angle E = 60^{\circ}\)