Question

in the figure below, ( mangle1=(x + 6)^circ ) and ( mangle2 = 5x^circ ). find the angle measures.

Explanation:

Step1: Set up the equation

Since \( \angle1\) and \( \angle2\) are complementary (they form a right - angle), \(m\angle1 + m\angle2=90^{\circ}\).
Substitute \(m\angle1=(x + 6)^{\circ}\) and \(m\angle2 = 5x^{\circ}\) into the equation:
\((x + 6)+5x=90\)

Step2: Simplify the equation

Combine like terms:
\(x+5x+6 = 90\)
\(6x+6=90\)

Step3: Solve for \(x\)

Subtract 6 from both sides:
\(6x+6 - 6=90 - 6\)
\(6x=84\)
Divide both sides by 6:
\(x=\frac{84}{6}=14\)

Step4: Find \(m\angle1\)

Substitute \(x = 14\) into \(m\angle1=(x + 6)^{\circ}\):
\(m\angle1=(14 + 6)^{\circ}=20^{\circ}\)

Step5: Find \(m\angle2\)

Substitute \(x = 14\) into \(m\angle2 = 5x^{\circ}\):
\(m\angle2=5\times14^{\circ}=70^{\circ}\)

Answer:

\(m\angle1 = 20^{\circ}\)
\(m\angle2 = 70^{\circ}\)