Question

in triangle abc, ( mangle b = (11x - 28)^circ ) and the measure of the exterior angle to ( angle b ) is ( (5x - 32)^circ ). find ( mangle b ).
options:
8°
16°
60°
137°

Explanation:

Step1: Use the exterior - angle theorem

The exterior - angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. Also, an interior angle and its exterior angle are supplementary. So, \(m\angle B+(5x - 32)^{\circ}=180^{\circ}\), which means \(m\angle B = 180-(5x - 32)= (212 - 5x)^{\circ}\). But we also know that \(m\angle B=(11x - 28)^{\circ}\).

Step2: Set up the equation

Set \(11x-28 = 212-5x\).
Add \(5x\) to both sides: \(11x + 5x-28=212-5x + 5x\), so \(16x-28 = 212\).
Add \(28\) to both sides: \(16x-28 + 28=212 + 28\), then \(16x=240\).
Divide both sides by \(16\): \(x=\frac{240}{16}=15\).

Step3: Find \(m\angle B\)

Substitute \(x = 15\) into \(m\angle B=(11x - 28)^{\circ}\).
\(m\angle B=(11\times15 - 28)^{\circ}=(165 - 28)^{\circ}=137^{\circ}\).

Answer:

\(137^{\circ}\)