Question

select the correct answer.
two exterior angles of a triangle measure 135° and 100°. what are the measures of the interior angles of the triangle?
a. 45°, 65°, 80°
b. 45°, 65°, 70°
c. 45°, 55°, 80°
d. 55°, 75°, 50°
e. 80°, 50°, 50°

Explanation:

Step1: Find first interior angle

An exterior angle and its adjacent interior angle are supplementary (sum to \(180^\circ\)). For the exterior angle \(135^\circ\), the interior angle is \(180^\circ - 135^\circ = 45^\circ\).

Step2: Find second interior angle

For the exterior angle \(100^\circ\), the interior angle is \(180^\circ - 100^\circ = 80^\circ\).

Step3: Find third interior angle

The sum of interior angles of a triangle is \(180^\circ\). Let the third angle be \(x\). Then \(45^\circ + 80^\circ + x = 180^\circ\). Solving for \(x\), we get \(x = 180^\circ - 45^\circ - 80^\circ = 55^\circ\)? Wait, no, wait: Wait, actually, when we have two exterior angles, we need to check which exterior angles they are. Wait, no, a triangle has three exterior angles, and the sum of exterior angles is \(360^\circ\). Wait, maybe I made a mistake. Let's correct:

The sum of exterior angles of any polygon is \(360^\circ\), so the third exterior angle is \(360^\circ - 135^\circ - 100^\circ = 125^\circ\). Then the interior angle adjacent to \(125^\circ\) is \(180^\circ - 125^\circ = 55^\circ\). Wait, but earlier, when I took the two exterior angles, maybe one of them is not adjacent to a non - overlapping interior angle? Wait, no, each exterior angle is adjacent to one interior angle (supplementary). But in a triangle, each interior angle has one exterior angle (adjacent, supplementary). But when we are given two exterior angles, we need to find the third exterior angle first, then find the interior angles.

Wait, let's start over:

1. Let the three exterior angles be \(E_1 = 135^\circ\), \(E_2 = 100^\circ\), and \(E_3\). Since the sum of exterior angles of a triangle is \(360^\circ\), \(E_3=360 - 135 - 100 = 125^\circ\).

1. Now, the interior angle \(I_1\) adjacent to \(E_1\) is \(180 - E_1=180 - 135 = 45^\circ\).

1. The interior angle \(I_2\) adjacent to \(E_2\) is \(180 - E_2 = 180 - 100 = 80^\circ\).

1. The interior angle \(I_3\) adjacent to \(E_3\) is \(180 - E_3=180 - 125 = 55^\circ\).

Wait, but looking at the options, option C is \(45^\circ, 55^\circ, 80^\circ\), which matches \(45^\circ\) (from \(135^\circ\) exterior), \(55^\circ\) (from \(125^\circ\) exterior), and \(80^\circ\) (from \(100^\circ\) exterior). Let's check the sum of interior angles: \(45 + 55+ 80 = 180\), which is correct.

Wait, earlier mistake was not considering that the two given exterior angles might not be adjacent to two non - overlapping interior angles in the sense that when we take two exterior angles, the third exterior angle is found by \(360 - 135 - 100 = 125\), then the interior angles are \(180 - 135 = 45\), \(180 - 100 = 80\), and \(180 - 125 = 55\). So the interior angles are \(45^\circ\), \(55^\circ\), \(80^\circ\), which is option C.

Answer:

C. \(45^\circ, 55^\circ, 80^\circ\)