Question

14
what is the value of y?
a. 130°
b. 50°
c. 25°
d. 65°

Explanation:

Step1: Identify the supplementary angle

The angle of \(130^\circ\) and the adjacent angle inside the triangle are supplementary, so the adjacent angle is \(180^\circ - 130^\circ = 50^\circ\).

Step2: Use the isosceles triangle property

The triangle is isosceles (two equal angles \(y\)), and the sum of angles in a triangle is \(180^\circ\). Let the two equal angles be \(y\), so \(y + y + 50^\circ = 180^\circ\).

Step3: Solve for \(y\)

Simplify the equation: \(2y = 180^\circ - 50^\circ = 130^\circ\), then \(y = \frac{130^\circ}{2} = 65^\circ\)? Wait, no, wait. Wait, no, the adjacent angle is \(50^\circ\)? Wait, no, wait, the exterior angle is \(130^\circ\), so the interior angle at that vertex is \(180 - 130 = 50\). Then the triangle has two angles \(y\) and one angle \(50\). So \(2y + 50 = 180\), so \(2y = 130\), \(y = 65\)? Wait, but the options have 65 as D? Wait, no, wait, maybe I made a mistake. Wait, no, the triangle is isosceles with two angles \(y\), so the exterior angle is equal to the sum of the two non - adjacent interior angles. Oh! Right, the exterior angle theorem: the exterior angle is equal to the sum of the two remote interior angles. So \(130^\circ = y + y\), so \(2y = 130^\circ\), so \(y = 65^\circ\)? Wait, no, that's not right. Wait, exterior angle theorem: the exterior angle is equal to the sum of the two non - adjacent interior angles. So if the exterior angle is \(130^\circ\), and the two non - adjacent interior angles are both \(y\), then \(130 = y + y\), so \(y = 65\). Wait, but let's check again. The sum of angles in a triangle: if the interior angle at the vertex with the exterior angle is \(180 - 130 = 50\), then the other two angles (both \(y\)) sum to \(180 - 50 = 130\), so each \(y\) is \(65\). So the answer is D? Wait, but the options: A. 130, B.50, C.25, D.65. So yes, D.

Answer:

D. \(65^\circ\)