Question

find the value of x.
5.
(3x + 6)°
81°
6.
(6x - 11)°
155°
7.
(8x)°
(10x - 32)°
8.
(4x + 60)°
(x + 15)°

Explanation:

Step1: Identify vertical - angle property

Vertical angles are equal. In problem 5, the angles \(81^{\circ}\) and \((3x + 6)^{\circ}\) are vertical angles. So we set up the equation \(3x+6 = 81\).
\[3x+6=81\]

Step2: Solve the equation for \(x\)

Subtract 6 from both sides: \(3x=81 - 6=75\). Then divide both sides by 3: \(x=\frac{75}{3}=25\).

In problem 6, the angles \(155^{\circ}\) and \((6x - 11)^{\circ}\) are vertical angles. Set up the equation \(6x-11 = 155\).

Step3: Solve the equation for \(x\)

Add 11 to both sides: \(6x=155 + 11=166\). Then divide both sides by 6: \(x=\frac{166}{6}=\frac{83}{3}\approx27.67\).

In problem 7, the angles \((8x)^{\circ}\) and \((10x - 32)^{\circ}\) are vertical angles. Set up the equation \(8x=10x - 32\).

Step4: Solve the equation for \(x\)

Subtract \(8x\) from both sides: \(0 = 10x-8x - 32\), which simplifies to \(2x-32 = 0\). Add 32 to both sides: \(2x=32\). Divide both sides by 2: \(x = 16\).

In problem 8, the angles \((4x + 60)^{\circ}\) and \((x + 15)^{\circ}\) are vertical angles. Set up the equation \(4x+60=x + 15\).

Step5: Solve the equation for \(x\)

Subtract \(x\) from both sides: \(4x-x+60=x - x+15\), which gives \(3x+60 = 15\). Subtract 60 from both sides: \(3x=15 - 60=-45\). Divide both sides by 3: \(x=-15\).

Answer:

1. \(x = 25\)
2. \(x=\frac{83}{3}\)
3. \(x = 16\)
4. \(x=-15\)