Question

vertical angles
find the value of x.
1)
2)
3)
4)
5)
6)

Explanation:

Step1: Use vertical angles property

Vertical angles are equal. In the first figure, \(6x = 30\).

Step2: Solve for \(x\)

Divide both sides by \(6\): \(x=\frac{30}{6}=5\).

Step3: Find \(m\angle MOL\)

\(m\angle MOL = 180 - 30=150^{\circ}\) (linear - pair with \(30^{\circ}\) angle).

Step4: For the second figure

Vertical angles: \(x - 17=180 - 157\).

Step5: Simplify the right - hand side

\(180 - 157 = 23\), so \(x-17 = 23\).

Step6: Solve for \(x\)

Add \(17\) to both sides: \(x=23 + 17=40\).
\(m\angle AOD\) and \(157^{\circ}\) are vertical angles, so \(m\angle AOD = 23^{\circ}\).

Step7: For the third figure

Vertical angles: \(8 + x=180 - 126\).

Step8: Simplify the right - hand side

\(180 - 126 = 54\), so \(8 + x=54\).

Step9: Solve for \(x\)

Subtract \(8\) from both sides: \(x=54 - 8 = 46\).
\(m\angle EOF\) and \(126^{\circ}\) are vertical angles, so \(m\angle EOF = 54^{\circ}\).

Step10: For the fourth figure

Vertical angles: \(\frac{x}{3}+45 = 180\) (linear - pair relationship for vertical - angle setup).
\(\frac{x}{3}=180 - 45\).

Step11: Simplify

\(\frac{x}{3}=135\).
Multiply both sides by \(3\): \(x = 405\).
\(m\angle SOR\) and \(\frac{x}{3}\) are vertical angles. Substitute \(x = 405\), \(\frac{405}{3}=135^{\circ}\), so \(m\angle SOR = 45^{\circ}\).

Step12: For the fifth figure

Vertical angles: \(x - 2=180 - 84\).

Step13: Simplify the right - hand side

\(180 - 84 = 96\), so \(x-2 = 96\).

Step14: Solve for \(x\)

Add \(2\) to both sides: \(x=96 + 2=98\).

Step15: For the sixth figure

Vertical angles: \(x + 11=180 - 108\).

Step16: Simplify the right - hand side

\(180 - 108 = 72\), so \(x + 11=72\).

Step17: Solve for \(x\)

Subtract \(11\) from both sides: \(x=72 - 11 = 61\).

Answer:

1. \(x = 5\), \(m\angle MOL=150^{\circ}\)
2. \(x = 40\), \(m\angle AOD = 23^{\circ}\)
3. \(x = 46\), \(m\angle EOF = 54^{\circ}\)
4. \(x = 405\), \(m\angle SOR = 45^{\circ}\)
5. \(x = 98\)
6. \(x = 61\)