Question

1. in each diagram, line k is parallel to line l, and line t intersects lines k and l.

based on the diagrams, which statement is true?

• the value of x is 79, because the two angles shown in each diagram are congruent.
• the value of x is 101, because the two angles shown in each diagram are supplementary.
• the value of x is greater than 90, because the two angles shown in each diagram are obtuse angles.
• the value of x is 11, because the two angles shown in each diagram are complementary.

Explanation:

Step1: Analyze angle relationships

In the first two diagrams, when parallel lines \(k\) and \(l\) are cut by transversal \(t\), the angles are congruent (113° and 113°, 127° and 127°), showing corresponding or vertical angles are equal. In the third, right angles (90°) are congruent. For the fourth, the angle with \(k\) is 79°, and \(x\) and 79° should be supplementary? Wait, no—wait, looking at the diagrams, the angles shown in each diagram (for \(k\) and \(l\) with transversal \(t\)): in first, 113° and 113° (congruent), second 127° and 127° (congruent), third 90° and 90° (congruent). Wait, no, wait the fourth diagram: the angle at \(k\) is 79°, and \(x\) and 79°—wait, actually, in each diagram, the two angles (one on \(k\), one on \(l\)) are congruent (corresponding angles, vertical angles, etc.). Wait, no, wait the fourth diagram: the angle with \(k\) is 79°, and \(x\) and 79°—wait, no, let's check the angle types. Wait, in the first diagram, 113° and 113°: congruent. Second, 127° and 127°: congruent. Third, 90° and 90°: congruent. So the pattern is that the two angles (one on \(k\), one on \(l\)) are congruent. So for the fourth diagram, the angle on \(k\) is 79°, so \(x\) should be equal? Wait no, wait the angle on \(k\) is 79°, and the angle \(x\) is adjacent? Wait no, maybe I misread. Wait, the transversal \(t\) intersects \(k\) and \(l\). The angle at \(k\) is 79°, and \(x\) is at \(l\). Wait, but in the first two diagrams, the angles are congruent (113° and 113°, 127° and 127°), so the relationship is congruent. Wait, but 79° and \(x\)—wait, no, maybe the angles are supplementary? Wait no, 113 + 113? No, 113 and 113 are equal. Wait, 127 and 127 are equal. 90 and 90 are equal. So the pattern is that the two angles (one on \(k\), one on \(l\)) are congruent. So for the fourth diagram, the angle on \(k\) is 79°, so \(x\) should be 79? Wait no, the options: first option says \(x = 79\) because the two angles are congruent. Wait, but wait, 79 and \(x\)—wait, maybe the angle at \(k\) is 79°, and \(x\) is equal to 79? Wait, no, wait the angle on \(k\) is 79°, and the angle \(x\) is at \(l\). Wait, maybe the angles are congruent (corresponding angles). So in each diagram, the two angles (one on \(k\), one on \(l\)) are congruent. So first diagram: 113° (on \(k\)) and 113° (on \(l\)): congruent. Second: 127° (on \(k\)) and 127° (on \(l\)): congruent. Third: 90° (on \(k\)) and 90° (on \(l\)): congruent. Fourth: 79° (on \(k\)) and \(x\) (on \(l\)): congruent. So \(x = 79\), because the two angles are congruent. Wait, but let's check the options. The first option: "The value of \(x\) is 79, because the two angles shown in each diagram are congruent." Let's verify the other options. Second option: \(x = 101\) (supplementary). But 79 + 101 = 180, but in the first diagrams, 113 + 113 = 226 ≠ 180, so supplementary is wrong. Third option: \(x > 90\) (obtuse). 79 is acute, so wrong. Fourth option: \(x = 11\) (complementary: 79 + 11 = 90, but first diagrams: 113 + 113 = 226 ≠ 90, so wrong. So the first option is correct: the two angles in each diagram are congruent (113°=113°, 127°=127°, 90°=90°, so 79°=x°), so \(x = 79\) because they are congruent.

Step2: Evaluate options

• Option 1: \(x = 79\) (congruent angles). Matches the pattern (113=113, 127=127, 90=90, so 79=x).
• Option 2: \(x = 101\) (supplementary). But 113+113≠180, so wrong.
• Option 3: \(x > 90\) (obtuse). 79 is acute, wrong.
• Option 4: \(x = 11\) (complementary). 113+113≠90, wrong.

Answer:

The value of \(x\) is 79, because the two angles shown in each diagram are congruent. (The first option, e.g., A. The value of \(x\) is 79, because the two angles shown in each diagram are congruent.)