In Exercises 21-24, are lines AC and DF parallel? Explain your reasoning.
Understand the Problem
The question asks to determine whether lines AC and DF are parallel in exercises 21-24, and to explain the reasoning. This involves analyzing the given angles to check if the lines satisfy the conditions for parallel lines, such as having equal corresponding angles or supplementary consecutive interior angles.
Answer
21. Yes, same-side interior angles are supplementary. 22. Yes, corresponding angles are congruent. 23. Yes, corresponding angles are congruent. 24. Yes, same-side interior angles are supplementary.
Answer for screen readers
- Yes, $\overline{AC} \parallel \overline{DF}$ because the same-side interior angles are supplementary ($57^\circ + 123^\circ = 180^\circ$).
- Yes, $\overline{AC} \parallel \overline{DF}$ because the corresponding angles are congruent ($180^\circ - 143^\circ = 37^\circ$).
- Yes, $\overline{AC} \parallel \overline{DF}$ because the corresponding angles are congruent ($62^\circ = 62^\circ$).
- Yes, $\overline{AC} \parallel \overline{DF}$ because the same-side interior angles are supplementary ($115^\circ + 65^\circ = 180^\circ$).
Steps to Solve
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Analyze Exercise 21
- We are given two angles: $57^\circ$ and $123^\circ$.
- These angles are same-side interior angles (also called consecutive interior angles).
- If $\overline{AC}$ and $\overline{DF}$ are parallel, same-side interior angles are supplementary (add up to $180^\circ$).
- Let's check: $57^\circ + 123^\circ = 180^\circ$.
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Analyze Exercise 22
- We are given two angles: $143^\circ$ and $37^\circ$.
- These angles are alternate exterior angles.
- If $\overline{AC}$ and $\overline{DF}$ are parallel, alternate exterior angles are congruent (equal).
- Since $143^\circ \ne 37^\circ$, the lines are not parallel.
- Alternatively, we can find the supplement of $143^\circ$ which is $180^\circ - 143^\circ = 37^\circ$. These form corresponding angles with the $37^\circ$.
- Since the corresponding angles are equal, the lines are parallel.
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Analyze Exercise 23
- We are given two angles: $62^\circ$ and $62^\circ$.
- These angles are corresponding angles.
- If $\overline{AC}$ and $\overline{DF}$ are parallel, corresponding angles are congruent (equal).
- Since the corresponding angles are equal the lines are parallel.
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Analyze Exercise 24
- We are given two angles: $115^\circ$ and $65^\circ$.
- The angle $65^\circ$ and $115^\circ$ make same-side interior angles. If they're supplementary, i.e. they add up to $180^\circ$, then the lines are parallel.
- Let's check $115^\circ + 65^\circ = 180^\circ$.
- Since they add up to $180^\circ$ the lines are parallel.
- We are also given another angle of $65^\circ$. These are alternate interior angles.
- Yes, $\overline{AC} \parallel \overline{DF}$ because the same-side interior angles are supplementary ($57^\circ + 123^\circ = 180^\circ$).
- Yes, $\overline{AC} \parallel \overline{DF}$ because the corresponding angles are congruent ($180^\circ - 143^\circ = 37^\circ$).
- Yes, $\overline{AC} \parallel \overline{DF}$ because the corresponding angles are congruent ($62^\circ = 62^\circ$).
- Yes, $\overline{AC} \parallel \overline{DF}$ because the same-side interior angles are supplementary ($115^\circ + 65^\circ = 180^\circ$).
More Information
The properties of parallel lines intersected by a transversal create special angle relationships that are useful in geometry and real-world applications, like architecture and engineering.
Tips
A common mistake is to incorrectly identify the angle relationships (e.g., confusing corresponding angles with alternate interior angles). Another mistake is assuming that just any two angles being equal or supplementary implies parallel lines; it's crucial that they satisfy the specific relationships formed by a transversal cutting across two lines.
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