Question

in the figure below, the ( mangle 6=(2x + 4)^{circ} ) and ( mangle 3=(7x + 5)^{circ} ). find what ( mangle 6= ).

Explanation:

Step1: Use the property of same - side interior angles

Since the two lines are parallel and cut by a transversal, \(\angle3\) and \(\angle6\) are same - side interior angles. So, \(m\angle3 + m\angle6=180^{\circ}\).
Substitute \(m\angle6=(2x + 4)^{\circ}\) and \(m\angle3=(7x + 5)^{\circ}\) into the equation: \((7x + 5)+(2x + 4)=180\).

Step2: Solve the equation for \(x\)

Combine like terms: \(7x+2x+5 + 4=180\), which simplifies to \(9x+9 = 180\).
Subtract 9 from both sides: \(9x=180 - 9=171\).
Divide both sides by 9: \(x=\frac{171}{9}=19\).

Step3: Find \(m\angle6\)

Substitute \(x = 19\) into \(m\angle6=(2x + 4)^{\circ}\).
\(m\angle6=(2\times19+4)^{\circ}=(38 + 4)^{\circ}=42^{\circ}\).

Answer:

\(42^{\circ}\)