Question

question 1 of 10
what is the value of y?
(there is a triangle with an exterior angle of 114° and a base angle of y°)
a. 33°
b. 66°
c. 114°
d. 57°

Explanation:

Step1: Identify the relationship

The angle of \(114^\circ\) and the angle adjacent to it (at the base of the triangle) are supplementary, so the adjacent angle is \(180^\circ - 114^\circ = 66^\circ\). But wait, actually, in the isosceles triangle (since two angles are \(y\)), the exterior angle (\(114^\circ\)) is equal to the sum of the two non - adjacent interior angles. But since the triangle has two equal angles (\(y\)), the exterior angle is equal to \(2y\)? Wait, no, let's correct. The angle adjacent to \(114^\circ\) is \(180 - 114=66^\circ\)? No, actually, the exterior angle theorem: the exterior angle is equal to the sum of the two remote interior angles. But in this triangle, if the base angles are \(y\) (since the triangle is isosceles, two angles are \(y\)), then the exterior angle \(114^\circ\) is equal to \(y + y=2y\)? Wait, no, the adjacent angle to \(114^\circ\) is \(180 - 114 = 66^\circ\), and in the triangle, the two base angles: wait, maybe the triangle is isosceles with two angles \(y\), and the exterior angle is equal to the sum of the two non - adjacent interior angles. Wait, no, let's start over.

The angle on the straight line with \(114^\circ\) is \(180 - 114=66^\circ\)? No, that's not right. Wait, the triangle has two angles labeled \(y\), so it's an isosceles triangle. The exterior angle ( \(114^\circ\)) is equal to the sum of the two non - adjacent interior angles. But the two non - adjacent interior angles are both \(y\)? Wait, no, the exterior angle is equal to the sum of the two remote interior angles. So if the exterior angle is \(114^\circ\), and the two remote interior angles are both \(y\) (since the triangle is isosceles), then \(114 = y + y\)? No, that would give \(y = 57\), but wait, no. Wait, the adjacent angle to \(114^\circ\) is \(180 - 114 = 66^\circ\), and in the triangle, the sum of angles in a triangle is \(180^\circ\). Wait, let's use the exterior angle theorem correctly. The exterior angle is equal to the sum of the two non - adjacent interior angles. So if the exterior angle is \(114^\circ\), and the two non - adjacent interior angles are the two angles of the triangle that are not adjacent to the exterior angle. If the triangle has angles \(y\), \(y\), and the third angle. Wait, no, the exterior angle is formed by extending one side of the triangle. So the angle adjacent to \(114^\circ\) is an interior angle of the triangle? Wait, no, the \(114^\circ\) is an exterior angle. So the interior angle adjacent to it is \(180 - 114 = 66^\circ\)? No, that's not. Wait, the triangle has two angles \(y\), so it's isosceles. The exterior angle \(114^\circ\) is equal to the sum of the two base angles (the two \(y\) angles). So \(114= y + y\)? No, that would be \(2y = 114\), so \(y = 57\)? Wait, no, that can't be. Wait, maybe the triangle has one angle equal to the angle adjacent to \(114^\circ\), and two angles \(y\). Wait, the angle adjacent to \(114^\circ\) is \(180 - 114 = 66^\circ\), and in the triangle, the sum of angles is \(180\). So \(66 + y + y=180\), so \(2y=180 - 66 = 114\), so \(y = 57\)? No, that's not. Wait, I'm getting confused. Let's use the exterior angle theorem properly. The exterior angle is equal to the sum of the two remote interior angles. So the exterior angle (\(114^\circ\)) is equal to the sum of the two interior angles that are not adjacent to it. If the triangle has two angles \(y\), then the exterior angle is equal to \(y + y\) (since those are the two remote interior angles). So \(114 = 2y\), so \(y=\frac{114}{2}=57\)? No, that's answer D. But wait, let's check again.

Wait, the angle on the straight line with \(114^\circ\) is \(180 - 114 = 66^\circ\). Then, in the triangle, the two base angles: if the triangle is isosceles with two angles \(y\), then the sum of angles in a triangle is \(180\). So \(66 + y + y=180\), so \(2y=180 - 66 = 114\), so \(y = 57\). Wait, that's correct. So the value of \(y\) is \(57^\circ\).

Step2: Verify

Using the exterior angle theorem: the exterior angle (\(114^\circ\)) is equal to the sum of the two non - adjacent interior angles. The two non - adjacent interior angles are both \(y\), so \(114 = y + y\), so \(y = 57^\circ\). Also, using the angle - sum property of a triangle: the angle adjacent to \(114^\circ\) is \(180 - 114 = 66^\circ\). Then, in the triangle, \(66 + y + y=180\), \(2y=114\), \(y = 57\).

Answer:

D. \(57^\circ\)