Question

find the measure of the angle indicated in bold.
13)
14)
15)
16)
find the measure of the indicated angle that makes lines u and v parallel.
17)
18)
19)
20)

Explanation:

Step1: Solve for \(x\) in problem 13

Since the angles \(x + 129\) and \(x+67\) are supplementary (they lie on the same - side of a transversal and between two parallel lines), we have the equation \((x + 129)+(x + 67)=180\).
Combine like - terms: \(2x+196 = 180\).
Subtract 196 from both sides: \(2x=180 - 196=-16\).
Divide by 2: \(x=-8\).
The angle \(x + 67=-8 + 67 = 59^{\circ}\).

Step2: Solve for \(x\) in problem 14

Since the angles \(12 + 16x\) and \(18x-4\) are supplementary (they lie on the same - side of a transversal and between two parallel lines), we have the equation \((12 + 16x)+(18x-4)=180\).
Combine like - terms: \(34x + 8=180\).
Subtract 8 from both sides: \(34x=180 - 8 = 172\).
Divide by 34: \(x=\frac{172}{34}=\frac{86}{17}\approx5.06\).
The angle \(18x-4=18\times\frac{86}{17}-4=\frac{1548 - 68}{17}=\frac{1480}{17}\approx87.06^{\circ}\).

Step3: Solve for \(x\) in problem 15

Since the angles \(10x\) and \(11x - 8\) are supplementary (they lie on the same - side of a transversal and between two parallel lines), we have the equation \(10x+(11x - 8)=180\).
Combine like - terms: \(21x-8 = 180\).
Add 8 to both sides: \(21x=188\).
Divide by 21: \(x=\frac{188}{21}\approx8.95\).
The angle \(10x = 10\times\frac{188}{21}=\frac{1880}{21}\approx89.52^{\circ}\).

Step4: Solve for \(x\) in problem 16

Since the angles \(x + 125\) and \(x+65\) are supplementary (they lie on the same - side of a transversal and between two parallel lines), we have the equation \((x + 125)+(x + 65)=180\).
Combine like - terms: \(2x+190 = 180\).
Subtract 190 from both sides: \(2x=-10\).
Divide by 2: \(x=-5\).
The angle \(x + 65=-5 + 65 = 60^{\circ}\).

Step5: Solve for the angle in problem 17

If lines \(u\) and \(v\) are parallel, the angle (let's call it \(y\)) and the \(132^{\circ}\) angle are supplementary (same - side interior angles). So \(y = 180-132=48^{\circ}\).

Step6: Solve for the angle in problem 18

If lines \(u\) and \(v\) are parallel, the angle (let's call it \(z\)) and the \(64^{\circ}\) angle are supplementary (same - side interior angles). So \(z = 180 - 64=116^{\circ}\).

Step7: Solve for the angle in problem 19

If lines \(u\) and \(v\) are parallel, the angle (let's call it \(w\)) and the \(130^{\circ}\) angle are supplementary (same - side interior angles). So \(w = 180-130 = 50^{\circ}\).

Step8: Solve for the angle in problem 20

If lines \(u\) and \(v\) are parallel, the angle (let's call it \(k\)) and the \(76^{\circ}\) angle are supplementary (same - side interior angles). So \(k = 180-76=104^{\circ}\).

Answer:

1. \(59^{\circ}\)
2. \(\frac{1480}{17}\approx87.06^{\circ}\)
3. \(\frac{1880}{21}\approx89.52^{\circ}\)
4. \(60^{\circ}\)
5. \(48^{\circ}\)
6. \(116^{\circ}\)
7. \(50^{\circ}\)
8. \(104^{\circ}\)