First we will decide which theorem or postulate from earlier in the chapter supports the conclusion in Theorem $3 - 9,$ and then we can do the same for the Perpendicular Transversal Theorem.

Theorem $3 - 9$

Let's review the content of Theorem $3 - 9 .$

Theorem $3 - 9$
In a plane, if two lines are perpendicular to the same line, then they are parallel each other.

Notice that the marked right angles are corresponding angles. Since they are both right angles, then they are congruent. Let's review the Converse of the Corresponding Angles Theorem.

Converse of the Corresponding Angles Theorem
If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel.

Therefore, by this theorem we can conclude that $m ∥ n .$

Perpendicular Transversal Theorem

Now we can review the content of the Perpendicular Transversal Theorem.

Perpendicular Transversal Theorem
In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other.

Notice that $∠ 1$ and $∠ 2$ are corresponding angles. Let's review the Corresponding Angles Theorem.

Corresponding Angles Theorem
If a transversal intersects two parallel lines, then corresponding angles are congruent.

Therefore, by this theorem we get that $∠ 1 ≅ ∠ 2 .$ We know that $∠ 2$ is a right angle, so we can conclude that $∠ 1$ is also a right angle. Thus, line $m$ is also perpendicular to line $n .$

Theorem 3-9

Perpendicular Transversal Theorem

Theorem $3 - 9$

Exercise