Math worksheet focusing on adjacent and vertical angles in two intersecting lines, with exercises and diagrams.

Problem Analysis:

The worksheet focuses on understanding adjacent angles and vertical angles formed by two intersecting lines. Let's solve the problems step by step.

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#### Part 1: Adjacent Angles
Adjacent angles are angles that share a common side and a common vertex but do not overlap.

1. Identify adjacent angles:
- From the diagram, we see that angles 1 and 2 are adjacent.
- Other pairs of adjacent angles are:
- ∠2 and ∠3
- ∠3 and ∠4
- ∠4 and ∠1

2. Special property of adjacent angles:
- Adjacent angles formed by two intersecting lines add up to 180° because they form a straight line.
- Therefore:
- ∠1 + ∠2 = 180°
- ∠2 + ∠3 = 180°
- ∠3 + ∠4 = 180°
- ∠4 + ∠1 = 180°

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#### Part 2: Vertical Angles
Vertical angles are angles opposite each other when two lines intersect. They are always congruent (equal in measure).

1. Identify vertical angles:
- From the diagram, ∠1 and ∠3 are vertical angles.
- Another pair of vertical angles is ∠2 and ∠4.

2. Relationship between vertical angles:
- Vertical angles are always equal in measure. For example, ∠1 = ∠3 and ∠2 = ∠4.

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#### Part 3: Classifying Angles
For each figure, determine whether the angles are adjacent, vertical, or neither.

1. Figure 1:
- The angles share a common side and a common vertex.
- Answer: Adjacent

2. Figure 2:
- The angles do not share a common side but are opposite each other.
- Answer: Vertical

3. Figure 3:
- The angles do not share a common side and are not opposite each other.
- Answer: Neither

4. Figure 4:
- The angles share a common vertex but do not share a common side.
- Answer: Neither

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#### Part 4: Solving for Angle Measures
We use the properties of adjacent and vertical angles to find the measures.

1. Figure 5:
- ∠1 and ∠2 are adjacent and form a straight line, so ∠1 + ∠2 = 180°.
- ∠2 and ∠3 are vertical angles, so ∠2 = ∠3.
- Let ∠1 = \( x \). Then ∠2 = 180° - \( x \).
- Since ∠2 = ∠3, ∠3 = 180° - \( x \).
- ∠1 + ∠2 = \( x + (180° - x) = 180° \)
- ∠2 + ∠3 = \( (180° - x) + (180° - x) = 360° - 2x \)

2. Figure 6:
- ∠1 and ∠3 are vertical angles, so ∠1 = ∠3.
- ∠2 and ∠4 are vertical angles, so ∠2 = ∠4.
- ∠1 + ∠2 = 180° (adjacent angles)
- Given ∠1 = 70°, then ∠2 = 180° - 70° = 110°.
- ∠3 = ∠1 = 70°
- ∠4 = ∠2 = 110°
- ∠1 + ∠2 + ∠3 = 70° + 110° + 70° = 250°

3. Figure 7:
- ∠1 and ∠2 are adjacent and form a straight line, so ∠1 + ∠2 = 180°.
- ∠2 and ∠3 are vertical angles, so ∠2 = ∠3.
- Given ∠2 = 125°, then ∠1 = 180° - 125° = 55°.
- ∠3 = ∠2 = 125°
- ∠2 + ∠3 = 125° + 125° = 250°

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#### Part 5: Drawing Supplementary Angles
Supplementary angles add up to 180°. To draw ∠3 vertical to ∠1:
1. Draw ∠1.
2. Draw ∠2 supplementary to ∠1 (so ∠1 + ∠2 = 180°).
3. Draw ∠3 vertical to ∠1 (so ∠3 = ∠1).

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Final Answers:

1. Adjacent angles: ∠2 and ∠3, ∠3 and ∠4, ∠4 and ∠1
2. Special property of adjacent angles: ∠1 + ∠2 = 180°, ∠2 + ∠3 = 180°, ∠3 + ∠4 = 180°, ∠4 + ∠1 = 180°
3. Vertical angles: ∠2 and ∠4
4. Classification of angles:
- Figure 1: Adjacent
- Figure 2: Vertical
- Figure 3: Neither
- Figure 4: Neither
5. Angle measures:
- Figure 5: ∠1 = \( x \), ∠2 = 180° - \( x \), ∠3 = 180° - \( x \)
- Figure 6: ∠1 = 70°, ∠2 = 110°, ∠3 = 70°, ∠4 = 110°, ∠1 + ∠2 + ∠3 = 250°
- Figure 7: ∠1 = 55°, ∠2 = 125°, ∠3 = 125°, ∠2 + ∠3 = 250°

Boxed Final Answer:

\[
\boxed{
\text{Adjacent angles: } \angle2 \text{ and } \angle3, \angle3 \text{ and } \angle4, \angle4 \text{ and } \angle1; \text{ Vertical angles: } \angle2 \text{ and } \angle4
}
\]