Question

1. find the value of the indicate variables.

Explanation:

Step1: Solve for \(x\) in the first diagram

Since the two angles \(83^{\circ}\) and \((8x + 1)^{\circ}\) are vertical angles, they are equal.
$$8x+1 = 83$$
Subtract \(1\) from both sides:
$$8x=83 - 1$$
$$8x=82$$
Divide both sides by \(8\):
$$x=\frac{82}{8}=\frac{41}{4}=10.25$$

Step2: Solve for \(x\) in the second diagram

The angles \(2(x + 15)^{\circ}\) and \((3x + 20)^{\circ}\) are supplementary (they form a linear - pair).
So, \(2(x + 15)+(3x + 20)=180\)
Expand the left - hand side: \(2x+30 + 3x+20 = 180\)
Combine like terms: \(5x+50 = 180\)
Subtract \(50\) from both sides: \(5x=180 - 50\)
$$5x=130$$
Divide both sides by \(5\): \(x = 26\)

Step3: Solve for \(x\) and \(y\) in the third diagram

First, for \(x\):
The angle \(100^{\circ}\) and \((2x + 80)^{\circ}\) are vertical angles.
$$2x+80 = 100$$
Subtract \(80\) from both sides: \(2x=100 - 80\)
$$2x=20$$
Divide both sides by \(2\): \(x = 10\)

Second, for \(y\):
The angle \(y+37\) and \(100^{\circ}\) are supplementary (they form a linear - pair).
$$y + 37+100=180$$
$$y+137 = 180$$
Subtract \(137\) from both sides: \(y=180 - 137\)
$$y = 43$$

Answer:

For the first diagram: \(x = 10.25\)
For the second diagram: \(x = 26\)
For the third diagram: \(x = 10,y = 43\)