Question

if ( fg = fi = 69 ), ( mangle fhg = p + 42^{circ} ), and ( mangle fhi = 4p ), what is ( mangle fhi )?
( mangle fhi=square^{circ} )

Explanation:

Step1: Use the Angle Bisector Theorem

Since \(FG = FI = 69\) and \(HG\perp FG\), \(HI\perp FI\), by the converse of the angle bisector theorem, \(FH\) is the angle bisector of \(\angle GHI\). So \(m\angle FHG=m\angle FHI\).

Step2: Set up the equation

We know that \(m\angle FHG = p + 42^{\circ}\) and \(m\angle FHI=4p\). Then \(p + 42=4p\).

Step3: Solve for \(p\)

Subtract \(p\) from both sides: \(42=4p - p\), so \(3p = 42\). Divide both sides by \(3\): \(p=\frac{42}{3}=14\).

Step4: Find \(m\angle FHI\)

Substitute \(p = 14\) into \(m\angle FHI = 4p\). Then \(m\angle FHI=4\times14 = 56^{\circ}\).

Answer:

\(56\)