404 Not Found - Free Printable

Let's solve each problem step by step based on the image you provided. The worksheet asks us to identify angle relationships in two categories:

1. Complementary, Supplementary, Vertical, or Adjacent
2. Alternate Interior, Corresponding, or Alternate Exterior

We'll go through each one and explain the reasoning.

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Part 1: Name the relationship: complementary, supplementary, vertical, or adjacent

#### 1)
- Two lines intersect, forming angles $ a $ and $ b $.
- These angles are next to each other and form a straight line (180°).
- They share a common side and vertex but do not overlap.
- Since they form a straight line, their sum is 180°.

✔ Answer: Supplementary

> Explanation: Angles that add up to 180° and form a straight line are supplementary.

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#### 2)
- Two rays form an angle with $ a $ and $ b $ sharing a vertex and a common side.
- They are next to each other and together make a larger angle.
- But there’s no indication they add to 90° or 180°.
- However, since they share a vertex and a side, and are side-by-side without overlapping, they are adjacent.

✔ Answer: Adjacent

> Explanation: Adjacent angles share a common vertex and side but do not overlap.

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#### 3)
- Two lines cross, forming four angles.
- $ a $ and $ b $ are opposite each other at the intersection.
- These are called vertical angles.

✔ Answer: Vertical

> Explanation: Vertical angles are formed when two lines intersect; they are opposite each other and equal in measure.

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#### 4)
- Two angles $ a $ and $ b $ meet at a point.
- There is a right-angle symbol at the corner where they meet.
- This means one of the angles is 90°, and together they form a right angle.
- So, $ a + b = 90^\circ $

✔ Answer: Complementary

> Explanation: Complementary angles add up to 90°.

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#### 5)
- A right angle is shown (square corner), with angles $ a $ and $ b $ forming it.
- So, $ a + b = 90^\circ $

✔ Answer: Complementary

> Explanation: Again, they form a right angle, so they are complementary.

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#### 6)
- Two angles $ a $ and $ b $ are next to each other, forming a straight line.
- They are adjacent and together make 180°.

✔ Answer: Supplementary

> Explanation: Adjacent angles that form a straight line are supplementary.

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Part 2: Name the relationship: alternate interior, corresponding, or alternate exterior

These involve parallel lines cut by a transversal.

#### 7)
- Two parallel lines are crossed by a transversal.
- Angle $ a $ and $ b $ are on opposite sides of the transversal and between the two lines.
- That fits the definition of alternate interior angles.

✔ Answer: Alternate Interior

> Explanation: Alternate interior angles are inside the parallel lines and on opposite sides of the transversal.

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#### 8)
- Parallel lines with a transversal.
- $ a $ and $ b $ are on the same side of the transversal.
- One is on the outside, one on the inside — wait, let's look carefully.
- Actually, both $ a $ and $ b $ are on the same side of the transversal and in corresponding positions relative to the parallel lines.

✔ Answer: Corresponding

> Explanation: Corresponding angles are in matching corners when a transversal crosses parallel lines.

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#### 9)
- Two parallel lines with a transversal.
- $ a $ and $ b $ are on opposite sides of the transversal.
- Both are outside the parallel lines.
- So, they are alternate exterior angles.

✔ Answer: Alternate Exterior

> Explanation: Alternate exterior angles are outside the parallel lines and on opposite sides of the transversal.

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#### 10)
- Two parallel lines with a transversal.
- $ a $ and $ b $ are on the same side of the transversal.
- One is above the top line, one is below the bottom line — so both are outside.
- They are on the same side of the transversal and outside → this is corresponding?

Wait — actually, $ a $ is above the top line, $ b $ is below the bottom line, and they are both on the right side of the transversal.

So they are in the same relative position — both are on the outer right side.

✔ Answer: Corresponding

> Explanation: Corresponding angles are in the same relative position at each intersection. Here, both are on the right side and outside the lines — so they correspond.

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1) | Supplementary |
| 2) | Adjacent |
| 3) | Vertical |
| 4) | Complementary |
| 5) | Complementary |
| 6) | Supplementary |
| 7) | Alternate Interior |
| 8) | Corresponding |
| 9) | Alternate Exterior |
| 10) | Corresponding |

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🔍 Summary of Definitions:

- Complementary: Add to 90°
- Supplementary: Add to 180°
- Vertical: Opposite angles formed by intersecting lines
- Adjacent: Share a side and vertex, no overlap
- Alternate Interior: Between lines, opposite sides of transversal
- Corresponding: Same position at each intersection
- Alternate Exterior: Outside lines, opposite sides of transversal

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