Question

write a two - column proof. given: m∠gfi = 138°. prove: m∠efi = 44°. statements: 1. m∠gfi = 138°; 2. m∠gfe + m∠efi = m∠gfi; 3. m∠gfe + m∠efi = blank°. reasons: 1. given; 2. angle addition postulate; 3. substitution property, (steps 1, 2)

Explanation:

Step1: Identify given angle measure

Given \(m\angle GFI = 138^{\circ}\)

Step2: Apply angle - addition postulate

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

Step3: Substitute the given value

Since \(m\angle GFI = 138^{\circ}\), substituting into the equation from Step 2 gives \(m\angle GFE + m\angle EFI=138^{\circ}\)

Step4: Express angles in terms of \(x\)

We know that \(m\angle GFE=(9x - 5)^{\circ}\) and \(m\angle EFI = 4x^{\circ}\), so \((9x-5)+4x=138\)

Step5: Combine like - terms

\(9x+4x-5 = 138\), which simplifies to \(13x-5=138\)

Step6: Solve for \(x\)

Add 5 to both sides: \(13x=138 + 5=143\). Then divide both sides by 13: \(x = 11\)

Step7: Find \(m\angle EFI\)

Since \(m\angle EFI = 4x^{\circ}\) and \(x = 11\), then \(m\angle EFI=4\times11^{\circ}=44^{\circ}\)

Answer:

The missing value in the table for Step 3 is \(138\) and we have proven that \(m\angle EFI = 44^{\circ}\)