Question

1. $m\angle1 =$

$m\angle2 =$

1. $m\angle1 =$

$m\angle2 =$

1. $m\angle1 =$

$m\angle2 =$

1. $m\angle1 =$

$m\angle2 =$

1. $m\angle1 =$

$m\angle2 =$

Explanation:

Step1: Use the property of vertical angles

Vertical angles are equal. For problem 10, the angle adjacent to \(120^{\circ}\) (let's call it \(x\)) satisfies \(x + 120^{\circ}=180^{\circ}\) (linear - pair). So \(x = 60^{\circ}\). Since \(\angle2\) and \(x\) are vertical angles, \(m\angle2 = 60^{\circ}\). And \(\angle1\) and \(120^{\circ}\) are vertical angles, so \(m\angle1=120^{\circ}\).

Step2: For problem 11

The angle adjacent to \(80^{\circ}\) (let's call it \(y\)) satisfies \(y + 80^{\circ}=180^{\circ}\) (linear - pair), so \(y = 100^{\circ}\). \(\angle1\) and \(y\) are vertical angles, \(m\angle1 = 100^{\circ}\). \(\angle2\) and \(80^{\circ}\) are vertical angles, so \(m\angle2=80^{\circ}\).

Step3: For problem 12

The angle adjacent to \(108^{\circ}\) (let's call it \(z\)) satisfies \(z + 108^{\circ}=180^{\circ}\) (linear - pair), so \(z = 72^{\circ}\). \(\angle2\) and \(z\) are vertical angles, \(m\angle2 = 72^{\circ}\). \(\angle1\) and \(108^{\circ}\) are vertical angles, so \(m\angle1=108^{\circ}\).

Step4: For problem 13

\(\angle1\) and \(56^{\circ}\) are vertical angles, so \(m\angle1 = 56^{\circ}\). The angle adjacent to \(56^{\circ}\) (let's call it \(w\)) satisfies \(w+56^{\circ}=180^{\circ}\) (linear - pair), \(w = 124^{\circ}\). \(\angle2\) and \(w\) are vertical angles, so \(m\angle2=124^{\circ}\).

Step5: For problem 14

The angle adjacent to \(117^{\circ}\) (let's call it \(u\)) satisfies \(u + 117^{\circ}=180^{\circ}\) (linear - pair), so \(u = 63^{\circ}\). \(\angle2\) and \(u\) are vertical angles, \(m\angle2 = 63^{\circ}\). \(\angle1\) and \(117^{\circ}\) are vertical angles, so \(m\angle1=117^{\circ}\).

Answer:

1. \(m\angle1 = 120^{\circ}\), \(m\angle2 = 60^{\circ}\)
2. \(m\angle1 = 100^{\circ}\), \(m\angle2 = 80^{\circ}\)
3. \(m\angle1 = 108^{\circ}\), \(m\angle2 = 72^{\circ}\)
4. \(m\angle1 = 56^{\circ}\), \(m\angle2 = 124^{\circ}\)
5. \(m\angle1 = 117^{\circ}\), \(m\angle2 = 63^{\circ}\)