Question

1. if ( mangle c = 49 ), find the values of ( x ) and ( y ). ( x = 49, y = 41 ) ( x = 90, y = 49 ) ( x = 41, y = 49 ) ( x = 90, y = 41 )

Explanation:

Step1: Use the property of isosceles triangle

Since \(AB = AC\) (marked with equal - side symbols), \(\triangle ABC\) is isosceles. \(AD\) is the median (because \(AB = AC\) and \(AD\) divides \(BC\) into two equal parts). In an isosceles triangle, the median from the vertex (\(A\)) to the base (\(BC\)) is also the altitude. So, \(\angle ADC=90^{\circ}\), which means \(x = 90\).

Step2: Use the angle - sum property of a triangle

In \(\triangle ADC\), we know that the sum of angles in a triangle is \(180^{\circ}\). Given \(\angle C = 49^{\circ}\) and \(\angle ADC=90^{\circ}\). Let \(\angle DAC=y\). Then, by the angle - sum formula \(\angle ADC+\angle C+\angle DAC = 180^{\circ}\). Substitute the values: \(90 + 49+y=180\).

$$y=180-(90 + 49)=180 - 139=41$$

Answer:

\(x = 90,y = 41\) (the fourth option)