Question

if m∠c = 66°, find the values of x and y.

x = 90°, y = 66°
x = 24°, y = 66°
x = 90°, y = 24°
x = 66°, y = 24°

Explanation:

Step1: Identify triangle type

Since \(AB = AC\), \(\triangle ABC\) is isosceles and \(AD\) is the perpendicular - bisector of \(BC\), so \(\angle ADB=\angle ADC = 90^{\circ}\), thus \(x = 90^{\circ}\).

Step2: Find angle \(y\)

In right - triangle \(ADC\), we know that \(\angle C=66^{\circ}\) and \(\angle ADC = 90^{\circ}\). Using the angle - sum property of a triangle (\(\angle A+\angle C+\angle ADC=180^{\circ}\) in \(\triangle ADC\)), we find \(\angle DAC=180^{\circ}-\angle ADC-\angle C\). Substituting the values, \(\angle DAC = 180^{\circ}-90^{\circ}-66^{\circ}=24^{\circ}\), so \(y = 24^{\circ}\).

Answer:

\(x = 90^{\circ},y = 24^{\circ}\) (corresponding to the third option)