Question

a side of the triangle below has been extended to form an exterior angle of 157°. find the value of x.
(there is a triangle image with an exterior angle of 157° and an interior angle of 109° and angle x°)
answer attempt 2 out of 2
x = | submit answer

Explanation:

Step1: Find adjacent interior angle

The exterior angle and its adjacent interior angle are supplementary, so the adjacent interior angle to \(157^\circ\) is \(180^\circ - 157^\circ = 23^\circ\).

Step2: Use triangle angle sum

The sum of angles in a triangle is \(180^\circ\). Let the third angle be \(x\). So \(x + 109^\circ + 23^\circ = 180^\circ\).

Step3: Solve for \(x\)

Simplify: \(x + 132^\circ = 180^\circ\). Then \(x = 180^\circ - 132^\circ = 48^\circ\)? Wait, no, wait. Wait, the adjacent angle to the exterior angle is \(180 - 157 = 23\), then the triangle has angles \(x\), \(109\), and \(23\)? Wait, no, maybe I messed up. Wait, the exterior angle theorem: the exterior angle is equal to the sum of the two non - adjacent interior angles. Wait, the exterior angle is \(157^\circ\), one non - adjacent angle is \(109^\circ\), and the other is \(x\). So by exterior angle theorem: \(157^\circ=x + 109^\circ\). Then \(x=157 - 109 = 48\)? Wait, no, wait, no. Wait, the exterior angle and the adjacent interior angle are supplementary. So adjacent interior angle is \(180 - 157 = 23\). Then in the triangle, angles are \(x\), \(109\), and \(23\). So \(x+109 + 23=180\), \(x + 132 = 180\), \(x = 48\). Wait, but let's check the exterior angle theorem. The exterior angle is equal to the sum of the two remote (non - adjacent) interior angles. So the exterior angle \(157^\circ\) should be equal to \(x+109^\circ\). So \(x = 157 - 109 = 48\). Yes, that's correct.

Answer:

\(x = 48\)