Question

kuta software - infinite geometry
the exterior angle theorem
find the measure of each angle indicated.
1)
2)
3)
4)
5)
6)
7)
8)
solve for x.
9)

Explanation:

Step1: Recall exterior - angle theorem

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

Step2: Solve problem 1

The exterior angle is 120°. One non - adjacent interior angle is 50°. Let the other non - adjacent interior angle be $\angle U$. Then $\angle U+50^{\circ}=120^{\circ}$, so $\angle U = 120^{\circ}-50^{\circ}=70^{\circ}$.

Step3: Solve problem 2

The exterior angle $\angle UTP = 115^{\circ}$, and one non - adjacent interior angle $\angle V = 50^{\circ}$. Let $\angle U$ be the other non - adjacent interior angle. Then $\angle U+50^{\circ}=115^{\circ}$, so $\angle U=115^{\circ}-50^{\circ}=65^{\circ}$.

Step4: Solve problem 3

The two non - adjacent interior angles are 70° and 50°. The exterior angle $\angle UTY=70^{\circ}+50^{\circ}=120^{\circ}$.

Step5: Solve problem 4

The two non - adjacent interior angles are 25° and 80°. The exterior angle $\angle PRT=25^{\circ}+80^{\circ}=105^{\circ}$.

Step6: Solve problem 5

The exterior angle $\angle ECT = 140^{\circ}$, and one non - adjacent interior angle is 45°. Let $\angle E$ be the other non - adjacent interior angle. Then $\angle E+45^{\circ}=140^{\circ}$, so $\angle E=140^{\circ}-45^{\circ}=95^{\circ}$.

Step7: Solve problem 6

The two non - adjacent interior angles are 80° and the angle adjacent to 110° (which is $180 - 110=70^{\circ}$). The exterior angle $\angle TUJ=80^{\circ}+70^{\circ}=150^{\circ}$.

Step8: Solve problem 7

The two non - adjacent interior angles are 28° and 58°. The exterior angle $\angle FGT=28^{\circ}+58^{\circ}=86^{\circ}$.

Step9: Solve problem 8

The two non - adjacent interior angles are 35° and 95°. The exterior angle $\angle GPQ=35^{\circ}+95^{\circ}=130^{\circ}$.

Step10: Solve problem 9

Using the exterior - angle theorem, we have $(4x + 2)+30=8 + 6x$.
First, simplify the left - hand side: $4x+32=8 + 6x$.
Then, move the x terms to one side: $32-8=6x - 4x$.
$24 = 2x$.
Divide both sides by 2: $x = 12$.

Answer:

1. $70^{\circ}$
2. $65^{\circ}$
3. $120^{\circ}$
4. $105^{\circ}$
5. $95^{\circ}$
6. $150^{\circ}$
7. $86^{\circ}$
8. $130^{\circ}$
9. $x = 12$