Question

1. if ( mangle c = 66^{circ} ), find the values of ( x ) and ( y ). ( x = 90^{circ}, y = 66^{circ} ) ( x = 90^{circ}, y = 24^{circ} ) ( x = 66^{circ}, y = 24^{circ} ) ( x = 24^{circ}, y = 66^{circ} )

Explanation:

Step1: Determine the value of \(x\)

Since \(AB = AC\) (marked with equal - length signs), \(\triangle ABC\) is isosceles. And \(AD\) is the median (because \(AB = AC\), the line from the vertex \(A\) to the mid - point \(D\) of \(BC\)). In an isosceles triangle, the median from the vertex to the base is also the altitude. So, \(AD\perp BC\), and \(x = 90^{\circ}\).

Step2: Calculate the value of \(y\)

In right - triangle \(ADC\), we know that the sum of the interior angles of a triangle is \(180^{\circ}\). Given \(\angle C=66^{\circ}\) and \(\angle ADC = x=90^{\circ}\).
Using the formula \(\angle CAD+y+\angle C+\angle ADC=180^{\circ}\) (in \(\triangle ADC\)), we can also use the fact that in right - triangle \(ADC\), \(y = 90^{\circ}-\angle C\).
Substitute \(\angle C = 66^{\circ}\) into the formula \(y=90^{\circ}-66^{\circ}\).
So, \(y = 24^{\circ}\).

Answer:

\(x = 90^{\circ},y = 24^{\circ}\) (the second option)