Question

write a two - column proof. given: m∠gfi = 127°. prove: m∠efi = 44°. 1. m∠gfi = 127° 2. m∠gfe + m∠efi = m∠gfi 3. m∠gfe + m∠efi = 127° 4. m∠gfe=(8x - 5)°, m∠efi = 4x° 5. 8x - 5+4x = 127

Explanation:

Step1: Given information

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

Step2: Angle - addition postulate

The sum of non - overlapping adjacent angles \(\angle GFE\) and \(\angle EFI\) is equal to \(\angle GFI\), so \(m\angle GFE + m\angle EFI=m\angle GFI\)

Step3: Substitute the value of \(m\angle GFI\)

Substitute \(m\angle GFI = 127^{\circ}\) into the equation from Step 2, getting \(m\angle GFE + m\angle EFI = 127^{\circ}\)

Step4: Given angle measures in terms of \(x\)

Given that \(m\angle GFE=(8x - 5)^{\circ}\) and \(m\angle EFI = 4x^{\circ}\)

Step5: Substitution

Substitute the expressions for \(m\angle GFE\) and \(m\angle EFI\) into the equation from Step 3. We get \((8x-5)+4x = 127\) (by the substitution property of equality)

Step6: Combine like - terms

\(8x+4x-5=127\), then \(12x-5 = 127\)

Step7: Use addition property of equality

Add 5 to both sides of the equation \(12x-5+5=127 + 5\), so \(12x=132\)

Step8: Use division property of equality

Divide both sides of the equation by 12: \(\frac{12x}{12}=\frac{132}{12}\), then \(x = 11\)

Step9: Find \(m\angle EFI\)

Since \(m\angle EFI = 4x^{\circ}\), substitute \(x = 11\) into the expression. So \(m\angle EFI=4\times11^{\circ}=44^{\circ}\)

Answer:

We have proven that \(m\angle EFI = 44^{\circ}\)