Question

practice describing the exterior angles of a triangle
study the example showing how to use the relationship between exterior and interior angles of a triangle. then solve problems 1 - 6.
example
find the value of x.
4x + 3x = 126
7x = 126
x = 18
1 what are the three exterior angle measures of the triangle in the example?
2 what is m∠cab? show your work.
3 can a triangle have an exterior angle that measures 90° at two different vertices? explain.

Explanation:

Step1: Recall exterior - angle property for question 1

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. In the example, one exterior angle is 126°. The other two exterior angles: Let the interior angles of the triangle be \(4x = 4\times18=72^{\circ}\) and \(3x = 3\times18 = 54^{\circ}\). The third interior angle is \(180-(72 + 54)=54^{\circ}\). The exterior angle corresponding to the \(72^{\circ}\) interior angle is \(180 - 72=108^{\circ}\), and the exterior angle corresponding to the \(54^{\circ}\) interior angle (not the 126° exterior - related one) is \(180 - 54 = 126^{\circ}\). So the three exterior - angle measures are 108°, 126°, 126°.

Step2: Use exterior - angle property for question 2

We know that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. For \(\triangle ABC\) with exterior angle \(\angle ACD=115^{\circ}\) and one non - adjacent interior angle \(\angle B = 80^{\circ}\). Let \(\angle CAB=x\). Then \(x+80 = 115\). Solving for \(x\): \(x=115 - 80\).
\(x = 35^{\circ}\), so \(m\angle CAB = 35^{\circ}\).

Step3: Analyze for question 3

If a triangle has an exterior angle of 90° at a vertex, the adjacent interior angle is \(180 - 90=90^{\circ}\). If there were two exterior angles of 90° at different vertices, then there would be two interior angles of 90° in the triangle. But the sum of the interior angles of a triangle is 180°. If there are two 90° angles, the sum of just those two angles is 180°, and there is no room for a third non - zero interior angle. So a triangle cannot have an exterior angle that measures 90° at two different vertices.

Answer:

1. 108°, 126°, 126°
2. \(m\angle CAB = 35^{\circ}\)
3. No. If a triangle has an exterior angle of 90° at a vertex, the adjacent interior angle is 90°. Having two such exterior angles would mean two 90° interior angles, which is impossible since the sum of interior angles of a triangle is 180°.