Question

finding angles ∠a and ∠b are complementary. find m∠a and m∠b.

1. ( mangle a = \frac{3}{4}x - 13 )

( mangle b = 3x - 17 )

Explanation:

Step1: Recall complementary angles property

Complementary angles sum to \(90^\circ\), so \(m\angle A + m\angle B = 90\).
Substitute \(m\angle A=\frac{3}{4}x - 13\) and \(m\angle B = 3x - 17\) into the equation:
\(\frac{3}{4}x - 13 + 3x - 17 = 90\)

Step2: Combine like terms

Combine the \(x\)-terms and constant terms:
\(\frac{3}{4}x+3x= \frac{3}{4}x+\frac{12}{4}x=\frac{15}{4}x\)
\(-13 - 17=-30\)
So the equation becomes \(\frac{15}{4}x - 30 = 90\)

Step3: Solve for \(x\)

Add 30 to both sides:
\(\frac{15}{4}x=90 + 30=120\)
Multiply both sides by \(\frac{4}{15}\):
\(x = 120\times\frac{4}{15}=\frac{480}{15}=32\)

Step4: Find \(m\angle A\)

Substitute \(x = 32\) into \(m\angle A=\frac{3}{4}x - 13\):
\(m\angle A=\frac{3}{4}(32)-13=24 - 13 = 11\)

Step5: Find \(m\angle B\)

Substitute \(x = 32\) into \(m\angle B = 3x - 17\):
\(m\angle B=3(32)-17 = 96 - 17 = 79\)

Answer:

\(m\angle A = 11^\circ\), \(m\angle B = 79^\circ\)