Question

find the value of x.
6.

in the exercises below, determine whether ∠f and ∠g are complementary, supplementary or neither.

1. m∠f = 137°, m∠g = 43°
2. m∠f = 87°, m∠g = 3°
3. m∠f = 22°, m∠g = 168°
4. the measure of ∠m is 43°. find the measure of a complement and a supplement of ∠m.

use the picture below to answer 11 - 13.

1. find the m∠1.
2. find the m∠2.
3. find the value of y.

Explanation:

Step1: Solve for x in question 6

Set up equation based on angle - sum property. If the two angles are adjacent and form a right - angle (assuming they are complementary), then \(53+(4x - 3)=90\).
\[

$$\begin{align*} 53+(4x - 3)&=90\\ 53 + 4x-3&=90\\ 50+4x&=90\\ 4x&=90 - 50\\ 4x&=40\\ x& = 10 \end{align*}$$

\]

Step2: Determine angle relationships in questions 7 - 9

Recall that complementary angles sum to \(90^{\circ}\) and supplementary angles sum to \(180^{\circ}\).

• For question 7: \(m\angle F+m\angle G=137 + 43=180^{\circ}\), so they are supplementary.
• For question 8: \(m\angle F+m\angle G=87+3 = 90^{\circ}\), so they are complementary.
• For question 9: \(m\angle F+m\angle G=22 + 168=190^{\circ}

eq90^{\circ}\) and \(190^{\circ}
eq180^{\circ}\), so they are neither.

Step3: Find complement and supplement of \(\angle M\) in question 10

The complement of \(\angle M\) is \(90 - 43=47^{\circ}\), and the supplement of \(\angle M\) is \(180 - 43 = 137^{\circ}\).

Step4: Solve for angles and y in questions 11 - 13

• Question 11: \(\angle1\) and the \(114^{\circ}\) angle are vertical angles, so \(m\angle1 = 114^{\circ}\).
• Question 12: \(\angle2\) and the \(114^{\circ}\) angle are supplementary. So \(m\angle2=180 - 114 = 66^{\circ}\).
• Question 13: Since \(\angle2=(2y + 12)^{\circ}\) and \(m\angle2 = 66^{\circ}\), we set up the equation \(2y+12=66\).

\[

$$\begin{align*} 2y+12&=66\\ 2y&=66 - 12\\ 2y&=54\\ y&=27 \end{align*}$$

\]

Answer:

1. \(x = 10\)
2. Supplementary
3. Complementary
4. Neither
5. Comp - \(47^{\circ}\), Supp - \(137^{\circ}\)
6. \(114^{\circ}\)
7. \(66^{\circ}\)
8. \(y = 27\)