Question

1. it is known that ∠efg and ∠gfh are a linear - angle pair. if m∠efg = 146°, then which of the following is the measure of ∠gfh? (1) 34° (2) 73° (3) 106° (4) 146° 4. if ∠e≅∠k, m∠e = 124°, and m∠k = 4x + 8, then which of the following is the value of x? (1) 7 (2) 12 (3) 23 (4) 29

Explanation:

Step1: Recall linear - angle pair property

The sum of angles in a linear - angle pair is 180°. Let \(m\angle EFG = 146^{\circ}\) and \(m\angle GFH=x\). Then \(m\angle EFG + m\angle GFH=180^{\circ}\).

Step2: Solve for \(m\angle GFH\)

\(x = 180^{\circ}-m\angle EFG\). Substitute \(m\angle EFG = 146^{\circ}\) into the equation: \(x=180 - 146=34^{\circ}\).

Step3: Use angle - equality property for second question

Since \(\angle E\cong\angle K\), then \(m\angle E=m\angle K\). Given \(m\angle E = 124^{\circ}\) and \(m\angle K = 4x + 8\), we set up the equation \(4x+8=124\).

Step4: Solve the equation for \(x\)

First, subtract 8 from both sides of the equation: \(4x=124 - 8=116\). Then divide both sides by 4: \(x=\frac{116}{4}=29\).

Answer:

1. (1) \(34^{\circ}\)
2. (4) 29