Question

2)
3)
4)
6)

Explanation:

1. First problem (top - left)

• Step1: Use the property of supplementary angles
• Supplementary angles sum to \(180^{\circ}\). So, \((x + 21)+147=180\).
• Simplify the equation: \(x+21 + 147=180\), then \(x+168 = 180\).
• Subtract \(168\) from both sides: \(x=180 - 168=12\).
• Step2: Find \(m\angle POS\)
• \(\angle POS\) and the angle of \(147^{\circ}\) are vertical angles. Vertical angles are equal. So \(m\angle POS = 147^{\circ}\)

2. Second problem (top - right)

• Step1: Use the property of vertical angles
• Vertical angles are equal. So \(2x=62\).
• Divide both sides by \(2\): \(x = 31\).
• Step2: Find \(m\angle MOL\)
• \(\angle MOL\) and the sum of \(\angle JOM\) and \(\angle KOL\) form a linear - pair (sum to \(180^{\circ}\)). Another way: \(\angle MOL=180-(62 + 62)=56^{\circ}\) (since \(\angle JOM=\angle KOL = 62^{\circ}\))

3. Third problem (middle - left)

• Step1: Use the property of supplementary angles
• \(\frac{x}{2}+95 = 180\) (supplementary angles).
• Subtract \(95\) from both sides: \(\frac{x}{2}=180 - 95 = 85\).
• Multiply both sides by \(2\): \(x = 170\).
• Step2: Find \(m\angle BOC\)
• \(\angle BOC\) and the angle of \(95^{\circ}\) are vertical angles. So \(m\angle BOC=95^{\circ}\)

4. Fourth problem (middle - right)

• Step1: Use the property of supplementary angles
• \(x-28+149 = 180\) (supplementary angles).
• Simplify the equation: \(x + 121=180\).
• Subtract \(121\) from both sides: \(x=180 - 121 = 59\).
• Step2: Find \(m\angle TOU\)
• \(\angle TOU\) and the angle of \(149^{\circ}\) are vertical angles. So \(m\angle TOU = 149^{\circ}\)

5. Fifth problem (bottom - left)

• Step1: Use the property of vertical angles
• The angle \((9 + x)^{\circ}\) and the angle adjacent to \(70^{\circ}\) (which is \(180 - 70=110^{\circ}\)) are vertical angles. So \(9+x=110\).
• Subtract \(9\) from both sides: \(x=110 - 9=101\)

6. Sixth problem (bottom - right)

• Step1: Use the property of supplementary angles
• \(x + 8+63=180\) (supplementary angles).
• Simplify the equation: \(x+71 = 180\).
• Subtract \(71\) from both sides: \(x=180 - 71 = 109\)

Answer:

1. \(x = 12\), \(m\angle POS=147^{\circ}\)
2. \(x = 31\), \(m\angle MOL = 56^{\circ}\)
3. \(x = 170\), \(m\angle BOC=95^{\circ}\)
4. \(x = 59\), \(m\angle TOU = 149^{\circ}\)
5. \(x = 101\)
6. \(x = 109\)