Question

identifying types of angles: check all relationships between ∠1 and ∠2.
1 diagram of two intersecting lines with ∠1 and ∠2 □ adjacent □ vertical □ complementary □ supplementary □ linear pair
2 diagram with a right angle □ adjacent □ vertical □ complementary □ supplementary □ linear pair
3 diagram of two intersecting lines □ adjacent □ vertical □ complementary □ supplementary □ linear pair
4 diagram of two angles with a common vertex □ adjacent □ vertical □ complementary □ supplementary □ linear pair
5 diagram with a right angle and intersecting lines □ adjacent □ vertical □ complementary □ supplementary □ linear pair
6 diagram with a right angle and intersecting lines □ adjacent □ vertical □ complementary □ supplementary □ linear pair

Explanation:

Problem 1:

Step1: Recall angle relationships. Vertical angles are opposite, formed by intersecting lines.

∠1 and ∠2 are vertical (opposite, from intersecting lines), supplementary (sum to 180° as linear pair? Wait, no—vertical angles are equal, but here they form a linear pair? Wait, no, two intersecting lines: ∠1 and ∠2 are vertical? Wait, no, in the first diagram, two lines intersect, so ∠1 and ∠2 are vertical? Wait, no, adjacent? No, vertical angles are opposite. Wait, no, when two lines intersect, vertical angles are equal, and linear pair is adjacent and supplementary. Wait, in diagram 1, ∠1 and ∠2: are they vertical? Yes, because they are opposite each other when two lines cross. Also, are they supplementary? No, vertical angles are equal, but if the lines are straight, a linear pair is supplementary. Wait, no—two intersecting lines form two pairs of vertical angles and two linear pairs. Wait, maybe I messed up. Let's recheck:

Vertical angles: opposite, equal. Linear pair: adjacent, form a straight line (supplementary, sum 180°). Adjacent: share a common side and vertex.

Diagram 1: two lines intersect. ∠1 and ∠2: do they share a side? No, they are opposite. So vertical. Are they supplementary? No, unless they are 90°, but here no. Wait, no—when two lines intersect, the adjacent angles are linear pairs (supplementary). So ∠1 and ∠2: vertical (yes), supplementary? No, linear pair? No, because they don't share a side. Wait, maybe the first diagram: ∠1 and ∠2 are vertical angles (so vertical is checked), and are they supplementary? No, unless the lines are perpendicular, but the diagram doesn't show that. Wait, maybe I made a mistake. Let's go step by step.

Step1: Identify angle types.

• Adjacent: share a common side and vertex. ∠1 and ∠2: no, they are opposite. So not adjacent.
• Vertical: opposite angles formed by intersecting lines. Yes, so vertical is checked.
• Complementary: sum to 90°: no info, so no.
• Supplementary: sum to 180°: no, unless linear pair. But they are not adjacent, so not supplementary.
• Linear Pair: adjacent and supplementary: no, not adjacent.

Wait, maybe the first diagram: two lines intersect, so ∠1 and ∠2 are vertical angles (so vertical is correct), and are they supplementary? No, vertical angles are equal, but if the lines are straight, the adjacent angles are supplementary. So ∠1 and ∠2: vertical (yes), and are they supplementary? No, unless they are 90°, but the diagram doesn't show that. Wait, maybe the first problem's ∠1 and ∠2: vertical (checked), and are they supplementary? No, linear pair? No. Wait, maybe I misread. Let's check the options again.

Wait, the first diagram: two lines intersect, so ∠1 and ∠2 are vertical angles (so vertical is a relationship), and are they supplementary? No, linear pair? No. Wait, maybe the answer is vertical.

Step1: Analyze the diagram. There's a right angle (90°) and ∠1, ∠2.

• Adjacent: share a common side and vertex. Yes, ∠1 and ∠2 share a side and vertex.
• Vertical: no, not opposite.
• Complementary: sum to 90° (since ∠1 + ∠2 + 90° = 180°? Wait, the straight line is 180°, with a right angle (90°), so ∠1 + ∠2 = 90°, so complementary.
• Supplementary: sum to 180°? No, ∠1 + ∠2 = 90°, so no.
• Linear Pair: adjacent and supplementary. ∠1 + ∠2 = 90° ≠ 180°, so no.

So relationships: Adjacent, Complementary.

Step1: Two lines intersect, ∠1 and ∠2.

• Adjacent: no, opposite.
• Vertical: yes, opposite angles.
• Complementary: no info, sum not 90°.
• Supplementary: no, unless linear pair. But they are vertical, so equal. Wait, no—when two lines intersect, vertical angles are equal, and adjacent angles are supplementary. So ∠1 and ∠2: vertical (yes), supplementary? No, linear pair? No. Wait, maybe the diagram: two lines intersect, so ∠1 and ∠2 are vertical angles (vertical), and are they supplementary? No, unless they are 90°, but diagram doesn't show. Wait, no—wait, in diagram 3, are ∠1 and ∠2 adjacent? No, opposite. So vertical (yes), and are they supplementary? No, linear pair? No. Wait, maybe the answer is vertical.

Wait, no—wait, two lines intersect, so ∠1 and ∠2 are vertical angles (vertical), and are they supplementary? No, but wait, maybe the diagram is different. Wait, diagram 3: two lines intersect, so ∠1 and ∠2 are vertical angles (so vertical is checked), and are they supplementary? No, linear pair? No. So vertical.

Answer:

(Problem 1): Vertical (check the box for Vertical)

Problem 2: