Question

what is the measure of all of the angles shown?

Explanation:

Step1: Find the measure of angle 4

Since angle 4 and the given \(120^{\circ}\) angle are adjacent and form a linear pair (sum to \(180^{\circ}\)), we have \(\angle4 = 180^{\circ}- 120^{\circ}=60^{\circ}\).

Step2: Find the measure of angle 1

Angle 1 and angle 4 are adjacent and form a linear pair. So \(\angle1=180^{\circ}-\angle4 = 180^{\circ}-60^{\circ}=120^{\circ}\).

Step3: Find the measure of angle 3

Angle 3 and the \(120^{\circ}\) angle are vertical angles. Vertical angles are equal. So \(\angle3 = 120^{\circ}\).

Step4: Find the measure of angle 2

Angle 2 and angle 4 are vertical angles. Vertical angles are equal. So \(\angle2=\angle4 = 60^{\circ}\).

Step5: Assume lines \(p\) and \(q\) are parallel (since it is a common problem - type with transversal \(t\))

Angle 4 and angle 8 are corresponding angles. If \(p\parallel q\), then \(\angle8=\angle4 = 60^{\circ}\).

Step6: Find the measure of angle 5

Angle 8 and angle 5 are adjacent and form a linear pair. So \(\angle5=180^{\circ}-\angle8=180^{\circ}-60^{\circ}=120^{\circ}\).

Step7: Find the measure of angle 7

Angle 7 and angle 5 are vertical angles. So \(\angle7=\angle5 = 120^{\circ}\).

Step8: Find the measure of angle 6

Angle 6 and angle 8 are vertical angles. So \(\angle6=\angle8 = 60^{\circ}\).

Answer:

\(\angle1 = 120^{\circ},\angle2 = 60^{\circ},\angle3 = 120^{\circ},\angle4 = 60^{\circ},\angle5 = 120^{\circ},\angle6 = 60^{\circ},\angle7 = 120^{\circ},\angle8 = 60^{\circ}\)