Question

write a two - column proof. given: m∠gfi = 141°. prove: m∠efi = 49°. statements: 1. m∠gfi = 141° (1 given). 2. m∠gfe + m∠efi = m∠gfi (2 angle addition postulate). 3. m∠gfe + m∠efi = blank° (3 substitution property, (steps 1, 2))

Explanation:

Step1: Recall given information

Given that \(m\angle GFI = 141^{\circ}\)

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 \(m\angle GFI = 141^{\circ}\), substituting into the equation from Step 2 gives \(m\angle GFE + m\angle EFI=141^{\circ}\)

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

We know that \(m\angle GFE=(8x - 3)^{\circ}\) and \(m\angle EFI = 4x^{\circ}\), so \((8x-3)+4x=141\)

Step5: Combine like - terms

\(8x+4x-3 = 141\), which simplifies to \(12x-3=141\)

Step6: Add 3 to both sides

\(12x-3 + 3=141 + 3\), so \(12x=144\)

Step7: Divide both sides by 12

\(\frac{12x}{12}=\frac{144}{12}\), then \(x = 12\)

Step8: Find \(m\angle EFI\)

Since \(m\angle EFI = 4x^{\circ}\), substituting \(x = 12\) gives \(m\angle EFI=4\times12^{\circ}=48^{\circ}\)

Answer:

\(m\angle EFI = 48^{\circ}\)