Question

1. ∠r and ∠s are complementary angles. if the m∠r=(12x - 3)° and m∠s=(7x - 2)°, find m∠r.

Explanation:

Step1: Recall the definition of complementary angles

Complementary angles sum to 90°. So, \(m\angle R + m\angle S=90^{\circ}\).

Step2: Substitute the given angle - expressions

Substitute \(m\angle R=(12x - 3)^{\circ}\) and \(m\angle S=(7x - 2)^{\circ}\) into the equation: \((12x - 3)+(7x - 2)=90\).

Step3: Simplify the left - hand side of the equation

Combine like terms: \(12x+7x-3 - 2 = 90\), which simplifies to \(19x-5 = 90\).

Step4: Solve for \(x\)

Add 5 to both sides of the equation: \(19x-5 + 5=90 + 5\), getting \(19x=95\). Then divide both sides by 19: \(x=\frac{95}{19}=5\).

Step5: Find \(m\angle R\)

Substitute \(x = 5\) into the expression for \(m\angle R\): \(m\angle R=(12x - 3)^{\circ}=(12\times5-3)^{\circ}=(60 - 3)^{\circ}=57^{\circ}\).

Answer:

\(57^{\circ}\)