Question

write a two - column proof. given: m∠gfi = 141°. prove: m∠efi = 48°. statements: 1. m∠gfi = 141°. reason: 1. given. 2. m∠gfe + m∠efi = m∠gfi. reason: 2. angle addition postulate. 3. m∠gfe + m∠efi = 141°. reason: 3. substitution property. (steps 1, 2). 4. m∠gfe=(8x - 3)°. m∠efi = 4x°. reason: 4. given. 5. 8x - 3+4x = 141. reason: 5.

Explanation:

Step1: Substitute angle measures

Since \(m\angle GFE=(8x - 3)^{\circ}\), \(m\angle EFI = 4x^{\circ}\) and \(m\angle GFE+m\angle EFI=141^{\circ}\), we substitute the angle - measure expressions into the equation. So, \((8x - 3)+4x=141\). The reason for this step is the Substitution Property (substituting the algebraic expressions for the angle measures into the angle - sum equation).

Step2: Combine like terms

Combine the \(x\) terms on the left - hand side of the equation: \((8x+4x)-3 = 141\), which simplifies to \(12x-3 = 141\).

Step3: Add 3 to both sides

Add 3 to both sides of the equation \(12x-3 = 141\) to isolate the term with \(x\). We get \(12x-3 + 3=141 + 3\), which simplifies to \(12x=144\). The reason is the Addition Property of Equality.

Step4: Solve for x

Divide both sides of the equation \(12x = 144\) by 12. So, \(\frac{12x}{12}=\frac{144}{12}\), and \(x = 12\).

Step5: Find \(m\angle EFI\)

Since \(m\angle EFI = 4x^{\circ}\), substitute \(x = 12\) into the expression. Then \(m\angle EFI=4\times12^{\circ}=48^{\circ}\).

Answer:

The reason for step 5 in the two - column proof is Substitution Property (substituting \(x = 12\) into the expression for \(m\angle EFI\)). And we have proved that \(m\angle EFI = 48^{\circ}\).