Question

1. if ( mangle bcd = 51^{circ} ), solve for ( x ).

Explanation:

Step1: Find the measure of ∠CFD

Since ∠CFD and the \(55^{\circ}\) angle are vertical angles, \(m\angle CFD = 55^{\circ}\).

Step2: Use the triangle angle - sum theorem in \(\triangle FCD\)

In \(\triangle FCD\), we know that \(m\angle FCD=51^{\circ}\) (given \(m\angle BCD = 51^{\circ}\)), \(m\angle CFD = 55^{\circ}\). By the triangle angle - sum theorem (\(m\angle FCD+m\angle CFD+m\angle FDC=180^{\circ}\)), we can find \(m\angle FDC\) as \(180-(51 + 55)=74^{\circ}\).

Step3: Use the property of alternate interior angles (assuming \(AB\parallel CD\))

If \(AB\parallel CD\), then \(14x + 4=74\) (alternate interior angles are equal).

Step4: Solve the equation for \(x\)

Subtract \(4\) from both sides of the equation \(14x+4 = 74\):
\(14x=74 - 4\)
\(14x=70\)
Divide both sides by \(14\): \(x=\frac{70}{14}=5\)

Answer:

\(x = 5\)