Question

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Explanation:

Step1: Solve Extra Practice 1

- **∠1 and ∠4**: Vertical angles? No, ∠1 and ∠3 are vertical, ∠2 and ∠4 are vertical. Wait, ∠1 and ∠4: supplementary? Wait, ∠4 is 135°, ∠1 and ∠4: adjacent? Wait, the lines intersect, so ∠1 and ∠3 are vertical, ∠2 and ∠4 are vertical. ∠1 and ∠2 are supplementary (linear pair). So:
- ∠2 and ∠4 are vertical angles, so \( m\angle2 = m\angle4 = 135^\circ \)
- ∠1 and ∠4: linear pair? Wait, ∠1 and ∠2 are linear pair, so \( m\angle1 + m\angle2 = 180^\circ \), so \( m\angle1 = 180 - 135 = 45^\circ \)
- ∠1 and ∠3 are vertical angles, so \( m\angle3 = m\angle1 = 45^\circ \)

Step2: Solve Extra Practice 2 (l || m, transversal n)

- **Vertical Angles**: ∠2 and ∠4, ∠5 and ∠7, ∠3 and ∠6, ∠1 and ∠8 are vertical. So \( m\angle2 = m\angle4 = m\angle5 = m\angle7 = 25^\circ \)
- **Linear Pairs**: ∠1 and ∠2, ∠3 and ∠4, ∠5 and ∠6, ∠7 and ∠8 are linear pairs (supplementary, sum to 180°). So \( m\angle1 = m\angle3 = m\angle6 = m\angle8 = 180 - 25 = 155^\circ \)
- **Angle Types**:
- ∠2 and ∠4: Vertical Angles (opposite, equal)
- ∠1 and ∠4: Linear Pair? No, ∠1 and ∠4: adjacent? Wait, ∠1 and ∠2 are linear pair, ∠2 and ∠4 are vertical. Wait, ∠1 and ∠4: vertical? No, ∠1 and ∠8 are vertical. Wait, ∠1 and ∠4: adjacent, form linear pair? Wait, ∠1 + ∠4 + ∠2 + ∠3? No, ∠1 and ∠4: adjacent, share a side, sum to 180? Wait, ∠1 and ∠2 are linear pair (180), ∠2 = 25, so ∠1 = 155. ∠4 = 25, so ∠1 and ∠4: supplementary? Wait, ∠1 + ∠4 = 155 + 25 = 180? Yes, linear pair? Wait, ∠1 and ∠4: adjacent, form a linear pair? Wait, the diagram: ∠1, ∠2, ∠3, ∠4 on line l; ∠5, ∠6, ∠7, ∠8 on line m. So ∠1 and ∠4: adjacent, linear pair? Wait, ∠1 and ∠2 are linear pair, ∠3 and ∠4 are linear pair. Wait, ∠1 and ∠4: vertical? No, ∠1 and ∠8 are vertical. Wait, maybe I messed up. Let's re-express:
- ∠2 (25°) and ∠4: vertical, so 25°
- ∠5 (25°) and ∠7: vertical, 25°
- ∠1 and ∠2: linear pair, so ∠1 = 180 - 25 = 155°
- ∠3 and ∠2: linear pair? No, ∠3 and ∠4: linear pair, so ∠3 = 180 - 25 = 155°
- ∠6 and ∠5: linear pair, so ∠6 = 180 - 25 = 155°
- ∠8 and ∠7: linear pair, so ∠8 = 180 - 25 = 155°
- **Angle Types**:
- ∠2 and ∠4: Vertical Angles
- ∠1 and ∠4: Linear Pair (supplementary, adjacent, form straight line)
- ∠2 and ∠5: Alternate Interior Angles? Wait, l || m, transversal n. ∠2 is on line l, ∠5 on line m, inside the lines, alternate sides: Alternate Interior Angles (equal, since l || m)
- ∠2 and ∠7: Corresponding Angles? ∠2 is on l, ∠7 on m, same position relative to transversal: ∠2 (top, right) and ∠7 (bottom, left)? No, ∠2 and ∠6: corresponding? Wait, ∠2 and ∠7: Alternate Exterior? No, ∠2 and ∠7: let's see, ∠2 (top, right of transversal) and ∠7 (bottom, left of transversal): maybe Vertical? No, ∠7 is 25°, ∠2 is 25°, so equal. Wait, ∠2 and ∠7: Alternate Interior? No, ∠2 is above l, ∠7 is below m. Wait, maybe ∠2 and ∠7: Vertical? No, ∠7 and ∠5 are vertical. Wait, ∠2 and ∠7: equal, so maybe Alternate Exterior? Wait, l || m, so ∠2 and ∠7: Alternate Exterior Angles (equal)
- ∠3 and ∠8: Alternate Interior Angles (l || m, so equal, 155°)

Step3: Solve Example 6 (a) (Vertical Angles)

- ∠1 and ∠2 are vertical angles? Wait, no, they are vertical angles (opposite, equal). Wait, the expressions: \( 3x + 20 \) and \( 10x - 15 \). Wait, vertical angles are equal? Wait, no, if they are vertical angles, then \( 3x + 20 = 10x - 15 \). Solve for x:
- \( 10x - 3x = 20 + 15 \)
- \( 7x = 35 \)
- \( x = 5 \)
- Then \( m\angle1 = 3(5) + 20 = 35^\circ \), \( m\angle2 = 10(5) - 15 = 35^\circ \) (wait, that's equal, so they are vertical angles)

Step4: Solve Example 6 (b) (Linear Pair)

- ∠3 and ∠4 are linear pair (supplementary), so \( (4n + 22) + (8n - 10) = 180 \)
- Combine like terms: \( 12n + 12 = 180 \)
- \( 12n = 168 \)
- \( n = 14 \)
- Then \( m\angle3 = 4(14) + 22 = 56 + 22 = 78^\circ \)
- \( m\angle4 = 8(14) - 10 = 112 - 10 = 102^\circ \) (check: 78 + 102 = 180, correct)

Answers (Summarized):

Extra Practice 1:

- \( m\angle1 = 45^\circ \)
- \( m\angle2 = 135^\circ \)
- \( m\angle3 = 45^\circ \)

Extra Practice 2:

- \( m\angle1 = 155^\circ \)
- \( m\angle2 = 25^\circ \)
- \( m\angle3 = 155^\circ \)
- \( m\angle4 = 25^\circ \)
- \( m\angle6 = 155^\circ \)
- \( m\angle7 = 25^\circ \)
- \( m\angle8 = 155^\circ \)
- Angle Types:
- ∠2 and ∠4: Vertical Angles
- ∠1 and ∠4: Linear Pair (supplementary)
- ∠2 and ∠5: Alternate Interior Angles (equal, since l || m)
- ∠2 and ∠7: Alternate Exterior Angles (equal, since l || m)
- ∠3 and ∠8: Alternate Interior Angles (equal, since l || m)

Example 6 (a):

- \( m\angle1 = 35^\circ \)
- \( m\angle2 = 35^\circ \)

Example 6 (b):

- \( m\angle3 = 78^\circ \)
- \( m\angle4 = 102^\circ \)

Answer:

s (Summarized):

Extra Practice 1:

• \( m\angle1 = 45^\circ \)
• \( m\angle2 = 135^\circ \)
• \( m\angle3 = 45^\circ \)

Extra Practice 2:

• \( m\angle1 = 155^\circ \)
• \( m\angle2 = 25^\circ \)
• \( m\angle3 = 155^\circ \)
• \( m\angle4 = 25^\circ \)
• \( m\angle6 = 155^\circ \)
• \( m\angle7 = 25^\circ \)
• \( m\angle8 = 155^\circ \)
• Angle Types:
• ∠2 and ∠4: Vertical Angles
• ∠1 and ∠4: Linear Pair (supplementary)
• ∠2 and ∠5: Alternate Interior Angles (equal, since l || m)
• ∠2 and ∠7: Alternate Exterior Angles (equal, since l || m)
• ∠3 and ∠8: Alternate Interior Angles (equal, since l || m)

Example 6 (a):

• \( m\angle1 = 35^\circ \)
• \( m\angle2 = 35^\circ \)

Example 6 (b):

• \( m\angle3 = 78^\circ \)
• \( m\angle4 = 102^\circ \)