The Perpendicular Transversal Theorem | Understanding the Relationship Between Perpendicular Transversals and Parallel Lines

In a plane, a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.

This statement is known as the Perpendicular Transversal Theorem

This statement is known as the Perpendicular Transversal Theorem.

First, let’s define a transversal. In geometry, a transversal is a line that intersects two or more other lines in a plane.

Parallel lines are lines in the same plane that never intersect. So, if we have two parallel lines and a transversal that intersects one of them, the Perpendicular Transversal Theorem states that if the transversal is perpendicular (forms a right angle) to one of the parallel lines, then it is also perpendicular to the other parallel line.

To prove this theorem, we can use the concept of alternate interior angles. When a transversal intersects parallel lines, it creates different types of angles, including alternate interior angles. Alternate interior angles are formed on opposite sides of the transversal and between the two parallel lines.

So, if the transversal is perpendicular to one of the parallel lines, one of the alternate interior angles formed will be a right angle. Since alternate interior angles that are formed by parallel lines are congruent (equal in measure), the alternate interior angle on the other side of the transversal will also be a right angle. This implies that the transversal is perpendicular to the other parallel line as well.

In summary, the Perpendicular Transversal Theorem states that if a transversal is perpendicular to one of two parallel lines, it is also perpendicular to the other parallel line.

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