Question

in the figure below, ( mangle1=(x + 46)^{circ} ) and ( mangle2 = 3x^{circ} ). find the angle measures.

Explanation:

Step1: Set up the equation

Since \(\angle1\) and \(\angle2\) are complementary (they form a right - angle, so \(m\angle1 + m\angle2=90^{\circ}\)). Substitute \(m\angle1=(x + 46)^{\circ}\) and \(m\angle2 = 3x^{\circ}\) into the equation: \((x + 46)+3x=90\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(x+3x + 46=90\), which gives \(4x+46 = 90\).

Step3: Solve for \(x\)

Subtract 46 from both sides: \(4x=90 - 46\), so \(4x=44\). Then divide both sides by 4: \(x=\frac{44}{4}=11\).

Step4: Find \(m\angle1\)

Substitute \(x = 11\) into \(m\angle1=(x + 46)^{\circ}\). Then \(m\angle1=(11 + 46)^{\circ}=57^{\circ}\).

Step5: Find \(m\angle2\)

Substitute \(x = 11\) into \(m\angle2=3x^{\circ}\). Then \(m\angle2=3\times11^{\circ}=33^{\circ}\).

Answer:

\(m\angle1 = 57^{\circ}\), \(m\angle2=33^{\circ}\)