Question

1. ∠3 is a complement of ∠4, and m∠3 = 46°. find m∠4.
2. ∠7 is a supplement of ∠8, and m∠7 = 109°. find m∠8.
3. find the measure of each angle
4. ∠efg and ∠lmn are supplementary angles. m∠efg=(3x + 17)°, and m∠lmn=($\frac{1}{2}$x - 5)°

Explanation:

Step1: Recall complement - angle formula

Complementary angles add up to 90°. So, \(m\angle3 + m\angle4=90^{\circ}\).

Step2: Solve for \(m\angle4\)

Given \(m\angle3 = 46^{\circ}\), then \(m\angle4=90^{\circ}-m\angle3\). Substitute \(m\angle3 = 46^{\circ}\) into the formula: \(m\angle4 = 90 - 46=44^{\circ}\).

Step3: Recall supplement - angle formula for \(\angle7\) and \(\angle8\)

Supplementary angles add up to 180°. So, \(m\angle7 + m\angle8=180^{\circ}\).

Step4: Solve for \(m\angle8\)

Given \(m\angle7 = 109^{\circ}\), then \(m\angle8=180^{\circ}-m\angle7\). Substitute \(m\angle7 = 109^{\circ}\) into the formula: \(m\angle8 = 180 - 109 = 71^{\circ}\).

Step5: For \(\angle BAC\) and \(\angle CAD\)

Since \(\angle BAC=(15x - 2)^{\circ}\) and \(\angle CAD=(7x + 4)^{\circ}\), and \(\angle BAD\) is a right - angle (\(90^{\circ}\)), then \((15x - 2)+(7x + 4)=90\).

Step6: Simplify the equation

Combine like terms: \(15x+7x-2 + 4=90\), which gives \(22x+2 = 90\).

Step7: Solve for \(x\)

Subtract 2 from both sides: \(22x=90 - 2=88\). Then divide both sides by 22: \(x=\frac{88}{22}=4\).

Step8: Find the measure of each angle

\(m\angle BAC=(15x - 2)^{\circ}\), substitute \(x = 4\): \(m\angle BAC=15\times4-2=60 - 2 = 58^{\circ}\). \(m\angle CAD=(7x + 4)^{\circ}\), substitute \(x = 4\): \(m\angle CAD=7\times4+4=28 + 4=32^{\circ}\).

Step9: For supplementary angles \(\angle EFG\) and \(\angle LMN\)

Since \(\angle EFG\) and \(\angle LMN\) are supplementary, \((3x + 17)+(\frac{1}{2}x-5)=180\).

Step10: Combine like terms

\(3x+\frac{1}{2}x+17 - 5=180\). First, get a common denominator for \(x\) terms: \(\frac{6x}{2}+\frac{1x}{2}+12 = 180\), which simplifies to \(\frac{7x}{2}+12 = 180\).

Step11: Solve for \(x\)

Subtract 12 from both sides: \(\frac{7x}{2}=180 - 12 = 168\). Multiply both sides by \(\frac{2}{7}\): \(x=\frac{168\times2}{7}=48\).

Step12: Find the measure of each angle

\(m\angle EFG=(3x + 17)^{\circ}\), substitute \(x = 48\): \(m\angle EFG=3\times48+17=144 + 17=161^{\circ}\). \(m\angle LMN=(\frac{1}{2}x - 5)^{\circ}\), substitute \(x = 48\): \(m\angle LMN=\frac{1}{2}\times48-5=24 - 5 = 19^{\circ}\).

Answer:

\(m\angle4 = 44^{\circ}\), \(m\angle8 = 71^{\circ}\), \(m\angle BAC = 58^{\circ}\), \(m\angle CAD = 32^{\circ}\), \(m\angle EFG = 161^{\circ}\), \(m\angle LMN = 19^{\circ}\)