Question

ii. vertical angles
find the value of x.
x = _____, m∠pos = _____
find the value of x.
x = _____, m∠mol = _____
find the value of x.
x = _____, m∠tou = _____
find the value of x.
x = _____, m∠lom = _____
find the value of x.
x = _____, m∠doe = _____

Explanation:

Step1: Recall vertical - angle property

Vertical angles are equal.

Step2: First problem

Since vertical angles are equal, we have \(x + 21=147\). Solving for \(x\):
\[x=147 - 21=126\]
\(\angle POS\) and the angle marked \(147^{\circ}\) are vertical angles, so \(m\angle POS = 147^{\circ}\).

Step3: Second problem

We know that \(62 = 2x\). Solving for \(x\):
\[x=\frac{62}{2}=31\]
\(\angle MOL\) and the \(62^{\circ}\) angle are vertical angles, so \(m\angle MOL = 62^{\circ}\).

Step4: Third problem

Since vertical angles are equal, \(x - 28=149\). Solving for \(x\):
\[x=149 + 28=177\]
\(\angle TOU\) and the \(149^{\circ}\) angle are vertical angles, so \(m\angle TOU = 149^{\circ}\).

Step5: Fourth problem

We have \(x+11 = 79\). Solving for \(x\):
\[x=79 - 11=68\]
\(\angle LOM\) and the \(79^{\circ}\) angle are vertical angles, so \(m\angle LOM = 79^{\circ}\).

Step6: Fifth problem

Since vertical angles are equal, \(x + 86=61\). Solving for \(x\):
\[x=61 - 86=- 25\]
\(\angle DOE\) and the \(61^{\circ}\) angle are vertical angles, so \(m\angle DOE = 61^{\circ}\).

Answer:

First problem: \(x = 126\), \(m\angle POS = 147^{\circ}\)
Second problem: \(x = 31\), \(m\angle MOL = 62^{\circ}\)
Third problem: \(x = 177\), \(m\angle TOU = 149^{\circ}\)
Fourth problem: \(x = 68\), \(m\angle LOM = 79^{\circ}\)
Fifth problem: \(x=-25\), \(m\angle DOE = 61^{\circ}\)