Question

use the diagram below to answer 23 - 27

a. diagram with 61° and 29° angles
b. diagram with angles 1, 2 and a right angle
c. diagram with angles 1, 2
d. diagram with angles 1, 2 and a right angle
e. diagram with 45° and 135° angles
f. diagram with angles 1, 2 and a right angle

1. which diagram(s) show adjacent angles? _______________
2. which diagram(s) show vertical angles? _______________
3. which diagram(s) show complementary angles? _______________
4. which diagram(s) show supplementary angles? _______________
5. which diagram(s) show a linear pair? _______________

Explanation:

23. Which diagram(s) show adjacent angles?

Adjacent angles share a common side and vertex, and no overlap.

• Diagram C: Angles 1 and 2 share a common vertex and side (the vertical ray), so adjacent.
• Diagram B: Angles 1 and 2? Wait, no—wait, Diagram B: the right angle is there, but angle 1 and 2? Wait, no, let's recheck. Wait, Diagram C: two angles with a common vertex and side (the vertical line), so adjacent. Diagram B: Wait, no, maybe I misread. Wait, Diagram C: angles 1 and 2 are adjacent (share a side and vertex). Diagram B: angle 1 and the right angle? No, maybe Diagram C, B? Wait, no, let's recall: Adjacent angles: same vertex, common side, no overlap.
• Diagram C: angles 1 and 2 share a vertex and the vertical side, so adjacent.
• Diagram B: angle 1 and the right angle? No, angle 1 and 2: do they share a side? The vertical line is common? Wait, maybe Diagram C, B, D? No, let's check each:
• A: two separate angles, no common side/vertex (different vertices? Or same? Wait, A has two angles, separate, so not adjacent.
• B: angle 1 and 2: do they share a side? The vertical line is a common side? Wait, the diagram B has a vertical line, a horizontal line (right angle), and a ray for angle 2. So angle 1 and 2: share the vertical line as a side, and common vertex. So adjacent? Wait, maybe B, C, D, F? No, maybe C, B, D? Wait, no, let's think again.

Wait, the correct adjacent angles:

• Diagram C: angles 1 and 2 share a common vertex and side (the vertical ray), so adjacent.
• Diagram B: angle 1 and the right angle? No, angle 1 and 2: share the vertical line as a side, common vertex. So adjacent.
• Diagram D: angle 1 and 2: share the vertical line? Wait, D has a vertical line, horizontal line (right angle), and a ray for angle 2. Angle 1 is between vertical and another ray. So angle 1 and 2: share the vertical line? Maybe.

Wait, maybe the answer is B, C, D, F? No, maybe I'm overcomplicating. Let's recall: Adjacent angles are next to each other, share a side and vertex.
So:

• C: yes (two angles with common vertex and side).
• B: angle 1 and 2: share vertical side, common vertex.
• D: angle 1 and 2: share vertical side, common vertex.
• F: angle 1 and 2: share vertical side? Wait, F has vertical line, horizontal line (right angle), and a ray for angle 1 and 2? Wait, maybe B, C, D, F? No, maybe the intended answer is B, C, D, F? Wait, no, let's check the diagrams again.

Wait, the problem is from a textbook, likely:

• Adjacent angles: C (angles 1 and 2 share a side), B (angle 1 and 2 share a side), D (angle 1 and 2 share a side), F (angle 1 and 2 share a side)? No, maybe C, B, D, F? Wait, maybe the answer is B, C, D, F? Wait, no, let's see:

Diagram C: two angles with a common vertex and a common side (the vertical ray), so adjacent.
Diagram B: angle 1 and 2: share the vertical line as a side, common vertex.
Diagram D: angle 1 and 2: share the vertical line as a side, common vertex.
Diagram F: angle 1 and 2: share the vertical line as a side, common vertex.
Wait, maybe the answer is B, C, D, F? But maybe I'm wrong. Alternatively, maybe C, B, D, F? Let's proceed.

Vertical angles are opposite angles formed by intersecting lines, equal in measure.

• Diagram D: angle 1 and 2 are vertical angles (formed by two intersecting lines, opposite each other).
• Diagram F: angle 1 and 2 are vertical angles (formed by two intersecting lines, opposite each other).
• Diagram B: angle 1 and 2? No, because there's a right angle, so maybe not. Wait, vertical angles are formed by two intersecting lines, so the lines must intersect. So D and F: in D, two lines intersect (vertical and the other), forming angle 1 and 2 as vertical. In F, same. B: has a right angle, so maybe the lines are not just intersecting, but with a right angle. Wait, vertical angles are formed when two lines intersect, so D and F: yes. B: angle 1 and 2: are they vertical? Let's see: in B, the vertical line and the other line (with the right angle) – no, maybe D and F.

Complementary angles sum to \( 90^\circ \).

• Diagram A: \( 61^\circ + 29^\circ = 90^\circ \), so complementary.
• Diagram C: if angles 1 and 2 sum to \( 90^\circ \), but we don't know, but A: 61 +29=90, so A is complementary.
• Diagram B: angle 1 and 2? If there's a right angle, maybe, but A: 61+29=90, so A.

Answer:

B, C, D, F (assuming these diagrams have adjacent angles; need to confirm, but based on adjacent angle definition: common side, common vertex, no overlap. So C (yes), B (yes), D (yes), F (yes); A and E: A has two separate angles, E has two separate angles.)

24. Which diagram(s) show vertical angles?