Question

in exercises 15–18, find the measure of the exterior angle. (see example 3.)
15.
16.
17.
18.

Explanation:

Problem 15

Step1: Recall the exterior angle theorem (the exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles).

For this triangle, the two non - adjacent interior angles are \(75^{\circ}\) and \(64^{\circ}\).

Step2: Calculate the measure of the exterior angle.

The measure of the exterior angle \(= 75^{\circ}+64^{\circ}\)
\(= 139^{\circ}\)

Step1: Recall the exterior angle theorem. The exterior angle \((2x - 2)^{\circ}\) is equal to the sum of the two non - adjacent interior angles \(x^{\circ}\) and \(45^{\circ}\). So we can set up the equation:

\(2x-2=x + 45\)

Step2: Solve the equation for \(x\).

Subtract \(x\) from both sides: \(2x-x-2=x - x+ 45\), which simplifies to \(x-2 = 45\)
Add 2 to both sides: \(x-2 + 2=45 + 2\), so \(x=47\)

Step3: Find the measure of the exterior angle.

Substitute \(x = 47\) into \((2x - 2)^{\circ}\): \(2\times47-2=94 - 2=92^{\circ}\)

Step1: Recall the exterior angle theorem. The exterior angle \((3x + 6)^{\circ}\) is equal to the sum of the two non - adjacent interior angles \(24^{\circ}\) and \((2x + 18)^{\circ}\). So we set up the equation:

\(3x+6=24+(2x + 18)\)

Step2: Simplify and solve the equation.

First, simplify the right - hand side: \(24+(2x + 18)=2x+42\)
So the equation becomes \(3x + 6=2x+42\)
Subtract \(2x\) from both sides: \(3x-2x + 6=2x-2x+42\), which gives \(x + 6=42\)
Subtract 6 from both sides: \(x+6 - 6=42 - 6\), so \(x = 36\)

Step3: Find the measure of the exterior angle.

Substitute \(x = 36\) into \((3x + 6)^{\circ}\): \(3\times36+6=108 + 6=114^{\circ}\)

Answer:

\(139^{\circ}\)

Problem 16