Question

write a two - column proof. given: m∠gfi = 127°. prove: m∠efi = 33°. statements reasons 1. m∠gfi = 127° 1. given 2. m∠gfe + m∠efi = m∠gfi 2. angle addition postulate 3. m∠gfe + m∠efi = ° 3. substitution property, (steps 1, 2)

Explanation:

Step1: Identify given angle

Given \(m\angle GFI = 127^{\circ}\).

Step2: Apply angle - addition postulate

By the Angle Addition Postulate, \(m\angle GFE+m\angle EFI=m\angle GFI\).

Step3: Substitute the given value

Substitute \(m\angle GFI = 127^{\circ}\) into the equation \(m\angle GFE + m\angle EFI=m\angle GFI\), so \(m\angle GFE+m\angle EFI = 127^{\circ}\).

Step4: Set up an equation using angle expressions

We know that \(m\angle GFE=(9x - 5)^{\circ}\) and \(m\angle EFI = 3x^{\circ}\), so \((9x-5)+3x=127\).

Step5: Combine like - terms

\(9x+3x-5 = 127\), which simplifies to \(12x-5 = 127\).

Step6: Solve for \(x\)

Add 5 to both sides: \(12x=127 + 5=132\). Then divide both sides by 12: \(x=\frac{132}{12}=11\).

Step7: Find \(m\angle EFI\)

Since \(m\angle EFI = 3x^{\circ}\), substitute \(x = 11\), so \(m\angle EFI=3\times11^{\circ}=33^{\circ}\).

Answer:

1. \(m\angle GFE + m\angle EFI=127^{\circ}\)