Question

1. if ∠cde is a straight angle. de bisects ∠gdh, m∠gde = (8x - 1)°, m∠edh = (6x + 15)°, and m∠cdf = 43°, find each measure.

Explanation:

Step1: Use the angle - bisector property

Since \(DE\) bisects \(\angle GDH\), we have \(m\angle GDE=m\angle EDH\).
So, \(8x - 1=6x + 15\).

Step2: Solve the equation for \(x\)

Subtract \(6x\) from both sides: \(8x-6x - 1=6x-6x + 15\), which gives \(2x-1 = 15\).
Add \(1\) to both sides: \(2x-1 + 1=15 + 1\), so \(2x=16\).
Divide both sides by \(2\): \(x=\frac{16}{2}=8\).

Step3: Find \(m\angle GDH\)

Since \(m\angle GDH=m\angle GDE + m\angle EDH\) and \(m\angle GDE=m\angle EDH=(8x - 1)=(6x + 15)\) (when \(x = 8\)), \(m\angle GDE=8\times8-1=63^{\circ}\), \(m\angle EDH=6\times8 + 15=63^{\circ}\). Then \(m\angle GDH=63^{\circ}+63^{\circ}=126^{\circ}\).

Step4: Find \(m\angle FDH\)

We know that \(\angle CDF+\angle FDH+\angle GDH = 180^{\circ}\) (because \(\angle CDE\) is a straight angle, \(m\angle CDE = 180^{\circ}\)). Given \(m\angle CDF = 43^{\circ}\) and \(m\angle GDH=126^{\circ}\).
So, \(m\angle FDH=180^{\circ}-43^{\circ}-126^{\circ}=11^{\circ}\).

Step5: Find \(m\angle FDE\)

\(m\angle FDE=m\angle FDH+m\angle EDH\). Substitute \(m\angle FDH = 11^{\circ}\) and \(m\angle EDH=63^{\circ}\). Then \(m\angle FDE=11^{\circ}+63^{\circ}=74^{\circ}\).

Answer:

\(x = 8\), \(m\angle GDH=126^{\circ}\), \(m\angle FDH = 11^{\circ}\), \(m\angle FDE=74^{\circ}\)