Question

find the measure of each angle indicated.
1)
2)
3)
4)
5)
6)
7)
8)

Explanation:

1)

Step1: Use triangle angle - sum formula

The sum of angles in a triangle is \(180^{\circ}\). Let the unknown angle be \(x\). Then \(x + 65^{\circ}+52^{\circ}=180^{\circ}\).

Step2: Solve for \(x\)

\(x=180^{\circ}-(65^{\circ} + 52^{\circ})=180^{\circ}-117^{\circ}=63^{\circ}\).

2)

Step1: Use right - triangle angle - sum formula

In a right - triangle (\(90^{\circ}\) angle), let the unknown angle be \(y\). Then \(y+40^{\circ}+90^{\circ}=180^{\circ}\).

Step2: Solve for \(y\)

\(y = 180^{\circ}-(40^{\circ}+90^{\circ})=180^{\circ}-130^{\circ}=50^{\circ}\).

3)

Step1: Use exterior - angle property (or triangle angle - sum)

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. Let the unknown angle be \(z\). Using the triangle angle - sum: The non - adjacent interior angle to \(130^{\circ}\) (let's call it \(a\)) is \(a = 180^{\circ}-130^{\circ}=50^{\circ}\). Then \(z+50^{\circ}+30^{\circ}=180^{\circ}\).

Step2: Solve for \(z\)

\(z=180^{\circ}-(50^{\circ}+30^{\circ})=100^{\circ}\).

4)

Step1: Use triangle angle - sum formula

Let the unknown angle be \(m\). Then \(m + 85^{\circ}+50^{\circ}=180^{\circ}\).

Step2: Solve for \(m\)

\(m=180^{\circ}-(85^{\circ}+50^{\circ})=45^{\circ}\).

5)

Step1: Find the interior angle adjacent to \(137^{\circ}\)

The interior angle \(n\) adjacent to \(137^{\circ}\) is \(n = 180^{\circ}-137^{\circ}=43^{\circ}\). Then, using the exterior - angle property (exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles). Let the unknown exterior angle be \(p\).

Step2: Calculate \(p\)

\(p=43^{\circ}+102^{\circ}=145^{\circ}\).

6)

Step1: Find the interior angle adjacent to \(35^{\circ}\)

The interior angle \(q\) adjacent to \(35^{\circ}\) is \(q = 180^{\circ}-35^{\circ}=145^{\circ}\). Then, using the exterior - angle property. Let the unknown angle be \(r\). The third interior angle of the triangle: \(s=180^{\circ}-(145^{\circ}+100^{\circ})=- 65^{\circ}\) (wrong, better use another approach). Using the fact that the sum of angles in a triangle: Let the triangle have angles \(35^{\circ}\) (vertical angle), \(100^{\circ}\), and the third angle \(t\). \(t = 180^{\circ}-(35^{\circ}+100^{\circ}) = 45^{\circ}\). Then the unknown angle \(r=180^{\circ}-45^{\circ}=135^{\circ}\).

7)

Step1: Use the exterior - angle property

Let the unknown angle be \(u\). The sum of the two non - adjacent interior angles ( \(30^{\circ}\) and \(20^{\circ}\)) gives the exterior angle.

Step2: Calculate \(u\)

\(u=30^{\circ}+20^{\circ}=50^{\circ}\).

8)

Step1: Find the interior angle adjacent to \(155^{\circ}\)

The interior angle \(v\) adjacent to \(155^{\circ}\) is \(v = 180^{\circ}-155^{\circ}=25^{\circ}\). Then, using the exterior - angle property. Let the unknown angle be \(w\).

Step2: Calculate \(w\)

\(w=25^{\circ}+40^{\circ}=65^{\circ}\).

Answer:

1. \(63^{\circ}\)
2. \(50^{\circ}\)
3. \(100^{\circ}\)
4. \(45^{\circ}\)
5. \(145^{\circ}\)
6. \(135^{\circ}\)
7. \(50^{\circ}\)
8. \(65^{\circ}\)