Question

1. review: find the value of the missing exterior angle.

11.
12.

Explanation:

Step1: Recall exterior - angle property of a triangle

The exterior - angle of a triangle is equal to the sum of the two non - adjacent interior angles.

For problem 10:

Let's consider the triangle. The exterior angle is \(x\), and the two non - adjacent interior angles are \(20^{\circ}\) and \(25^{\circ}\).
By the exterior - angle property of a triangle, \(x=20 + 25\).

$$x = 45^{\circ}$$

For problem 11:

The exterior angle is \(3x\), and the two non - adjacent interior angles are \((x + 15)^{\circ}\) and \(43^{\circ}\).
By the exterior - angle property, \(3x=(x + 15)+43\).

Step2: Solve the equation for \(x\)

$$3x=x + 15+43$$

$$3x=x + 58$$

Subtract \(x\) from both sides: \(3x−x=x + 58−x\).

$$2x=58$$

Divide both sides by 2: \(x = 29\).
The exterior angle \(3x=3\times29 = 87^{\circ}\)

For problem 12:

The sum of the interior angles of the polygon and the exterior angles around it.
The sum of the exterior angles of any polygon is \(360^{\circ}\).
The given exterior angles are \(70^{\circ}\), \(36^{\circ}\), \(82^{\circ}\), and two angles of measure \(x\) each.
So, \(2x+70 + 36+82=360\).

$$2x+188 = 360$$

Step2: Solve for \(x\)

Subtract 188 from both sides: \(2x=360 - 188\).

$$2x=172$$

Divide both sides by 2: \(x = 86^{\circ}\)

Answer:

1. \(x = 45^{\circ}\)
2. The missing exterior angle is \(87^{\circ}\)
3. \(x = 86^{\circ}\)