Question

in the diagram below, line m is parallel to line n, and line t intersects lines m and n. based on the diagram, which of the following statements is true? the value of x is 251, because the measures of the angles shown in the diagram add up to 360° and 251 + 109 = 360. the value of x is 71, because the measures of the angles shown in the diagram add up to 180°. the value of x is 71, because the angles shown in the diagram are obtuse angles. the value of x is 109, because the angles shown in the diagram are obtuse angles.

Explanation:

Step1: Recall angle - relationship of parallel lines

When a transversal intersects two parallel lines, corresponding angles are equal. The angle of \(109^{\circ}\) and the angle adjacent to \(x^{\circ}\) are corresponding angles, so the angle adjacent to \(x^{\circ}\) is \(109^{\circ}\).

Step2: Use linear - pair property

Since \(x^{\circ}\) and the \(109^{\circ}\) - angle form a linear pair (a straight - line), and the sum of angles in a linear pair is \(180^{\circ}\). We have the equation \(x + 109=180\).

Step3: Solve for \(x\)

Subtract 109 from both sides of the equation \(x + 109 = 180\), so \(x=180 - 109=71\).

Answer:

The value of \(x\) is 71, because the measures of the angles shown in the diagram add up to \(180^{\circ}\).