Question

1. in each diagram, line m is parallel to line n, and line t intersects lines m and n. the value of x is less than 90°, because the two angles shown in each diagram are acute angles. the value of x is 133, because the two angles shown in each diagram are supplementary. the value of x is 43°, because the two angles shown in each diagram are complimentary. the value of x is 47°, because the two angles shown in each diagram are congruent.

Explanation:

Step1: Recall angle - relationships for parallel lines

When a transversal \(t\) intersects two parallel lines \(m\) and \(n\), corresponding angles are congruent, alternate - interior angles are congruent, and same - side interior angles are supplementary.

Step2: Analyze the given diagrams

In the first diagram, if we assume the angle of \(125^{\circ}\) and the angle adjacent to \(x\) are corresponding or alternate - interior angles, and since linear pairs of angles sum to \(180^{\circ}\), if the non - \(x\) angle is \(125^{\circ}\), then \(x = 180^{\circ}-125^{\circ}=55^{\circ}\). In the second diagram, if the non - \(x\) angle is \(105^{\circ}\), then \(x = 180^{\circ}-105^{\circ}=75^{\circ}\). In the third diagram, if the non - \(x\) angle is \(47^{\circ}\), then \(x = 180^{\circ}-47^{\circ}=133^{\circ}\). In the fourth diagram, since the vertical angles are congruent and the lines are parallel, we use the angle - relationship rules. The angles shown in each diagram are supplementary (as they are same - side interior angles for the parallel lines \(m\) and \(n\) intersected by the transversal \(t\)).

Answer:

The value of \(x\) is 133, because the two angles shown in each diagram are supplementary.