Question

1. m∠1 = 51°

m∠2 = 34°
m∠3 = 95°
m∠4 = 86°
m∠5 = 47°
m∠6 = 159°
m∠7 = 59°

Explanation:

Step1: Use linear - pair property for ∠1

A linear - pair of angles sums to 180°. Since the angle adjacent to ∠1 is 129°, then \(m\angle1=180^{\circ}- 129^{\circ}=51^{\circ}\).

Step2: Use linear - pair property for ∠2

The angle adjacent to ∠2 and the 95° angle form a linear - pair. First, find the angle adjacent to ∠2. Let it be \(x\), then \(x = 180^{\circ}-95^{\circ}=85^{\circ}\). Then, since the sum of angles in a triangle with ∠1 and ∠2 is 180°, and we know ∠1 = 51°. So \(m\angle2=180^{\circ}-(51^{\circ}+95^{\circ}) = 34^{\circ}\).

Step3: Use linear - pair property for ∠3

The angle adjacent to ∠3 and the 47° angle form a linear - pair. Let the adjacent angle be \(y\), then \(y=180^{\circ}-47^{\circ}=133^{\circ}\). Also, considering the angles around the intersection point, \(m\angle3 = 180^{\circ}-(34^{\circ}+51^{\circ})=95^{\circ}\).

Step4: Use angle - sum property of a triangle for ∠4

In the triangle with ∠3, ∠4 and the 47° angle, \(m\angle4=180^{\circ}-(95^{\circ}+47^{\circ}) = 38^{\circ}\).

Step5: ∠5 is vertical to the 47° angle

Vertical angles are equal, so \(m\angle5 = 47^{\circ}\).

Step6: Use linear - pair property for ∠7

The angle adjacent to ∠7 is 121°. So \(m\angle7=180^{\circ}-121^{\circ}=59^{\circ}\).

Step7: Use angle - sum property of a triangle for ∠6

In the triangle with ∠5 and ∠7, \(m\angle6=180^{\circ}-(47^{\circ}+59^{\circ})=74^{\circ}\).

Answer:

\(m\angle1 = 51^{\circ}\), \(m\angle2 = 34^{\circ}\), \(m\angle3 = 95^{\circ}\), \(m\angle4 = 38^{\circ}\), \(m\angle5 = 47^{\circ}\), \(m\angle6 = 74^{\circ}\), \(m\angle7 = 59^{\circ}\)