Question

a drafting table is shown in the image. what are the measures of the vertical angles formed by the intersection of the legs? (12x - 78)° (4x + 42)° find the value of x that makes the vertical angles congruent.

Explanation:

Step1: Set up equation based on vertical - angle property

Vertical angles are congruent. So, we set \(12x - 78=4x + 42\).

Step2: Isolate the variable terms

Subtract \(4x\) from both sides: \(12x-4x - 78=4x-4x + 42\), which simplifies to \(8x-78 = 42\).

Step3: Isolate the variable

Add 78 to both sides: \(8x-78 + 78=42 + 78\), getting \(8x=120\).

Step4: Solve for \(x\)

Divide both sides by 8: \(x=\frac{120}{8}=15\).

Step5: Find the measure of the vertical angles

Substitute \(x = 15\) into either expression for the vertical - angle measure. Using \(4x + 42\), we have \(4\times15+42=60 + 42=102^{\circ}\).

Answer:

\(x = 15\), and the measure of each vertical angle is \(102^{\circ}\)