Question

in the figure below, ( m angle 1=(x + 42)^{circ} ) and ( m angle 2=2 x^{circ} ). find the angle measures.

Explanation:

Step1: Set up the equation

Since \(\angle1+\angle2 = 90^{\circ}\) (complementary angles), and \(m\angle1=(x + 42)^{\circ}\), \(m\angle2 = 2x^{\circ}\), we have \((x + 42)+2x=90\).

Step2: Simplify the left - hand side

Combine like terms: \(x+2x + 42=90\), which gives \(3x+42 = 90\).

Step3: Solve for \(x\)

Subtract 42 from both sides: \(3x=90 - 42\), so \(3x=48\). Then divide both sides by 3: \(x=\frac{48}{3}=16\).

Step4: Find \(m\angle1\)

Substitute \(x = 16\) into \(m\angle1=(x + 42)^{\circ}\): \(m\angle1=(16 + 42)^{\circ}=58^{\circ}\).

Step5: Find \(m\angle2\)

Substitute \(x = 16\) into \(m\angle2 = 2x^{\circ}\): \(m\angle2=2\times16^{\circ}=32^{\circ}\).

Answer:

\(m\angle1 = 58^{\circ}\), \(m\angle2=32^{\circ}\)