Question

given: ∠4 is complementary to ∠5. the m∠4=(x + 5)° and m∠5=(15x - 11)°, find the measure of the smaller angle.

Explanation:

Step1: Use the definition of complementary angles

Complementary angles sum to \(90^{\circ}\). So, \((x + 5)+(15x-11)=90\).

Step2: Simplify the equation

Combine like - terms: \(x+15x + 5-11=90\), which simplifies to \(16x-6 = 90\).

Step3: Solve for \(x\)

Add \(6\) to both sides: \(16x=90 + 6=96\). Then divide by \(16\): \(x=\frac{96}{16}=6\).

Step4: Find the measures of \(\angle4\) and \(\angle5\)

For \(\angle4\): \(m\angle4=(x + 5)^{\circ}\). Substitute \(x = 6\), so \(m\angle4=(6 + 5)^{\circ}=11^{\circ}\).
For \(\angle5\): \(m\angle5=(15x-11)^{\circ}\). Substitute \(x = 6\), so \(m\angle5=(15\times6-11)^{\circ}=(90 - 11)^{\circ}=79^{\circ}\).

Answer:

\(11^{\circ}\)