Question

chapter 3: angle pairs & perpendicular lines
find the measure of each angle below.

1. (17) ∠abc

(b)15. (18) ∠bef
answer each question below.

1. (15) the measure of an angle is 5 times as great as the measure of its complement. find the measure of the angle.

(c)17. (16) the measure of an angle is 8 greater than 3 times its supplement. find the measure of the angle.

Explanation:

Step1: Find \(x\) using vertical - angles

Vertical angles are equal. So, \(x + y=96\) and \(2x=y\). Substitute \(y = 2x\) into \(x + y=96\), we get \(x+2x=96\), which simplifies to \(3x = 96\). Then \(x=\frac{96}{3}=32\).

Step2: Find \(y\)

Since \(y = 2x\) and \(x = 32\), then \(y=2\times32 = 64\).

Step3: Find \(\angle ABC\)

\(\angle ABC=(x + y)^{\circ}\), and since \(x + y=96\), \(\angle ABC = 96^{\circ}\).

Step4: Find \(\angle BEF\)

\(\angle BEF=y^{\circ}\), and since \(y = 64\), \(\angle BEF=64^{\circ}\).

Step5: Solve for the angle in question 16

Let the angle be \(a\) and its complement be \(90 - a\). Given \(a = 5(90 - a)\). Expand: \(a=450-5a\). Add \(5a\) to both sides: \(a + 5a=450\), \(6a=450\), \(a = 75^{\circ}\).

Step6: Solve for the angle in question 17

Let the angle be \(b\) and its supplement be \(180 - b\). Given \(b=3(180 - b)+8\). Expand: \(b = 540-3b+8\). Add \(3b\) to both sides: \(b+3b=540 + 8\), \(4b=548\), \(b = 137^{\circ}\).

Answer:

1. \(\angle ABC = 96^{\circ}\)
2. \(\angle BEF=64^{\circ}\)
3. The measure of the angle is \(75^{\circ}\)
4. The measure of the angle is \(137^{\circ}\)