Question

solve the following word problem using supplementary or complementary angle definitions. find the value of x and the measure of ∠a and ∠b. angles a and b are complementary. if m∠a = 5x - 10 and m∠b = 7x + 4.

Explanation:

Step1: Use complementary angles sum formula

Complementary angles sum to \(90^{\circ}\). So, \(m\angle A + m\angle B=90\). Substitute \(m\angle A = 5x - 10\) and \(m\angle B = 7x + 4\) into the equation: \((5x - 10)+(7x + 4)=90\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(5x+7x-10 + 4=90\), which simplifies to \(12x-6 = 90\).

Step3: Solve for \(x\)

Add \(6\) to both sides: \(12x-6+6=90 + 6\), so \(12x=96\). Then divide both sides by \(12\): \(x=\frac{96}{12}=8\).

Step4: Find \(m\angle A\)

Substitute \(x = 8\) into \(m\angle A=5x - 10\). \(m\angle A=5\times8-10=40 - 10=30\).

Step5: Find \(m\angle B\)

Substitute \(x = 8\) into \(m\angle B=7x + 4\). \(m\angle B=7\times8+4=56 + 4=60\).

Answer:

\(x = 8\), \(\angle A=30\), \(\angle B = 60\)