Question

1. if m∠def=(7x + 4)°, m∠deg=(5x + 1)°, and m∠gef = 23°, find each measure.

Explanation:

Step1: Set up angle - addition equation

Since \(m\angle DEF=m\angle DEG + m\angle GEF\), we have \((7x + 4)=(5x + 1)+23\).

Step2: Simplify the right - hand side

\((5x + 1)+23=5x+24\), so the equation becomes \(7x + 4=5x+24\).

Step3: Subtract \(5x\) from both sides

\(7x-5x + 4=5x-5x+24\), which simplifies to \(2x+4 = 24\).

Step4: Subtract 4 from both sides

\(2x+4 - 4=24 - 4\), giving \(2x=20\).

Step5: Solve for \(x\)

Divide both sides by 2: \(\frac{2x}{2}=\frac{20}{2}\), so \(x = 10\).

Step6: Find \(m\angle DEG\)

Substitute \(x = 10\) into the expression for \(m\angle DEG\): \(m\angle DEG=(5x + 1)=(5\times10+1)=51^{\circ}\).

Step7: Find \(m\angle DEF\)

Substitute \(x = 10\) into the expression for \(m\angle DEF\): \(m\angle DEF=(7x + 4)=(7\times10+4)=74^{\circ}\).

Answer:

\(x = 10\)
\(m\angle DEG=51^{\circ}\)
\(m\angle DEF=74^{\circ}\)