To determine which fact would allow you to prove that lines \( l \) and \( m \) are parallel, we need to use the properties of angles formed when a transversal crosses two lines. Here are the properties we typically use:
1. **Alternate Interior Angles**: If alternate interior angles are equal, then the lines are parallel.
2. **Corresponding Angles**: If corresponding angles are equal, then the lines are parallel.
3. **Consecutive Interior Angles**: If consecutive (same-side) interior angles sum to 180°, then the lines are parallel.
Let's analyze each fact in the context of these properties:
1. **\( m∠2 + m∠4 = 180° \)**: This statement relates to consecutive interior angles. If the sum of consecutive interior angles is 180°, then the lines are parallel by the Consecutive Interior Angles Theorem.
2. **\( m∠1 = m∠8 \)**: This statement suggests that the corresponding angles are equal. If corresponding angles are equal, the lines are parallel.
3. **\( m∠6 + m∠3 = 180° \)**: This statement relates to consecutive interior angles. If the sum of consecutive interior angles is 180°, then the lines are parallel by the Consecutive Interior Angles Theorem.
4. **\( m∠7 = m∠6 \)**: This statement does not correspond to any of the standard angle relationships (alternate interior, corresponding, or consecutive interior) used to prove lines are parallel. Therefore, it does not help prove that lines \( l \) and \( m \) are parallel.
Given these analyses, the facts that help to prove lines \( l \) and \( m \) are parallel are:
- \( m∠2 + m∠4 = 180° \)
- \( m∠1 = m∠8 \)
- \( m∠6 + m∠3 = 180° \)
Since we are asked to select one of these options, and all the valid conditions (Facts 1, 2, and 3) can justify the parallelism, any one of \( m∠2 + m∠4 = 180° \), \( m∠1 = m∠8 \), or \( m∠6 + m∠3 = 180° \) would suffice. Thus:
**The most straightforward answer based on the given choices is:**
**\( m∠1 = m∠8 \)** (Corresponding Angles Theorem).
#14 Which of the following facts, if true, would allow you to prove that lines l and m are parallel?
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m∠2 + m∠4 = 180°
m∠1 = m∠8
m∠6 + m∠3 = 180°
m∠7 = m∠6
1 answer