Question

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

Explanation:

Step1: Use the exterior angle theorem

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

Step2: Calculate for problem 1

For problem 1:
The exterior angle at \(V\) is \(120^{\circ}\). Let the unknown angle at \(U\) be \(x\).
By the exterior angle theorem \(x + 50^{\circ}=120^{\circ}\)
\(x=120^{\circ}- 50^{\circ}=70^{\circ}\)

Step3: Calculate for problem 2

For problem 2:
The exterior angle at \(T\) is \(115^{\circ}\). Let the unknown angle at \(U\) be \(y\).
First, find the angle adjacent to \(115^{\circ}\) inside the triangle: \(180^{\circ}-115^{\circ} = 65^{\circ}\)
Using the angle - sum property of a triangle (\(V + U+\text{adjacent angle}=180^{\circ}\)), \(50^{\circ}+y + 65^{\circ}=180^{\circ}\)
\(y=180^{\circ}-(50^{\circ}+65^{\circ})=65^{\circ}\)

Step4: Calculate for problem 3

For problem 3:
Using the angle - sum property of a triangle (\(S + T+ \text{angle at }U\text{ inside the triangle}=180^{\circ}\)), the angle inside the triangle at \(U\) is \(180^{\circ}-(70^{\circ}+50^{\circ}) = 60^{\circ}\)
The unknown angle (let's call it \(z\)) and the \(60^{\circ}\) angle inside the triangle are supplementary (\(z+60^{\circ}=180^{\circ}\))
\(z = 120^{\circ}\)

Step5: Calculate for problem 4

For problem 4:
First, find the angle at \(T\) inside the triangle. Since the angle adjacent to \(80^{\circ}\) is \(100^{\circ}\) (because \(180^{\circ}-80^{\circ}=100^{\circ}\))
Using the exterior angle theorem (the unknown angle \(a\) is an exterior angle of the small triangle), \(a=25^{\circ}+100^{\circ}=125^{\circ}\)

Step6: Calculate for problem 5

For problem 5:
The angle adjacent to \(140^{\circ}\) inside the triangle is \(40^{\circ}\) (\(180^{\circ}-140^{\circ}=40^{\circ}\))
Using the angle - sum property of a triangle (\(E + D+\text{adjacent angle}=180^{\circ}\)), \(E+45^{\circ}+40^{\circ}=180^{\circ}\)
\(E = 95^{\circ}\)

Step7: Calculate for problem 6

For problem 6:
The angle adjacent to \(110^{\circ}\) inside the triangle is \(70^{\circ}\) (\(180^{\circ}-110^{\circ}=70^{\circ}\))
Using the angle - sum property of a triangle (\(T+\text{adjacent angle}+ \text{angle at }U\text{ inside the triangle}=180^{\circ}\)), let the unknown angle at \(U\) be \(b\)
\(80^{\circ}+70^{\circ}+b = 180^{\circ}\)
\(b=30^{\circ}\)

Step8: Calculate for problem 7

For problem 7:
Using the exterior angle theorem, the unknown angle \(c\) is \(26^{\circ}+58^{\circ}=84^{\circ}\)

Step9: Calculate for problem 8

For problem 8:
First, find the angle inside the triangle adjacent to \(93^{\circ}\), which is \(87^{\circ}\) (\(180^{\circ}-93^{\circ}=87^{\circ}\))
Using the exterior angle theorem, the unknown angle \(d\) is \(35^{\circ}+87^{\circ}=122^{\circ}\)

Answer:

1. \(70^{\circ}\)
2. \(65^{\circ}\)
3. \(120^{\circ}\)
4. \(125^{\circ}\)
5. \(95^{\circ}\)
6. \(30^{\circ}\)
7. \(84^{\circ}\)
8. \(122^{\circ}\)