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Chapter 10: Problem 72

If a transversal is perpendicular to one of two parallel lines, must it be perpendicular to the other parallel line as well? Explain your answer.

Short Answer

Expert verified
Yes, if a transversal is perpendicular to one of two parallel lines, it does have to be perpendicular to the other parallel line. This is due to the fact that corresponding angles between a transversal and parallel lines are equal, hence if it forms a 90-degree angle with one line it will form a 90-degree (perpendicular) angle with the other line as well.

Step by step solution

01

Understanding parallel lines and transversals

Parallel lines are two lines in a plane that do not intersect or touch each other at any point. A transversal is a line that cuts across other lines, possibly intersecting them. In this case, the transversal intersects two parallel lines.
02

Analyzing the property of corresponding angles

When a transversal intersects two parallel lines, it forms corresponding angles that are equal. So, if the transversal is perpendicular to one parallel line, it forms a 90-degree angle which with that line. Consequently, the corresponding angle formed with the other parallel line is also 90 degrees.
03

Drawing the conclusion

Due to the identical nature of corresponding angles, the transversal must also be perpendicular to the second line. This is because the measure of a right angle (or a perpendicular) is 90 degrees. Therefore, if a transversal is perpendicular to one of two parallel lines, it must be perpendicular to the other parallel line as well.

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