Perpendicular Transversal Theorem

Introduction of Perpendicular Transversal

A transversal line connects two or more (typically parallel) lines. A perpendicular transversal is one line that crosses parallel lines at right angles. The corresponding angles around each intersection are identical in measuring if the transversal cuts over parallel lines.

Perpendicular Transversal Theorem

According to the perpendicular transversal theorem, if two parallel lines exist in the same plane and a line is perpendicular to one of them, it is likewise perpendicular to the other.

Consider two parallel lines, L1 and L2, as well as a perpendicular line k to L1.

We already know that,

Angles 1 and 2 form a pair of corresponding angles on the same side and are thus identical because L1 and L2 are parallel to each other, and k is a transversal.

As a result, line k is perpendicular to lines L1 and L2. This establishes the perpendicular transversal theorem, which asserts that if two parallel lines intersect and another line is perpendicular to one of them, it is also perpendicular to the other. 

Perpendicular Transversal Theorem Converse

In a plane, the converse perpendicular transversal theorem states that if two lines are perpendicular to each other, they are parallel. These theorems can be used to show that two angles are congruent or two lines are parallel.

When a transversal intersects two lines, alternate interior angles, alternate exterior angles, and corresponding angles are all congruent. The opposite of the theorem also holds. If two related angles are congruent, the transversal must cut two parallel lines. The lines are parallel if two alternate interior or alternate exterior angles are congruent. 

How to Prove the Perpendicular Transversal Theorem?

So, if we have two potentially parallel lines that are intersected by a transversal, can we use the converse of the parallel lines theorem? Is it enough to argue that these two lines are parallel if we have alternate inner angles or alternate exterior angles, or congruent and corresponding angles? Yes, it is, as we can see right here. So, if two lines and a transversal create alternate interior angles, which are congruent, it is sufficient to conclude that these two lines must be parallel.

If you have many outer perspectives. If these two alternate external angles are congruent, you don't even need to know about the inside ones. So it suffices to say that they are parallel.

Finally, there are equivalent angles. These two lines must be parallel if they have one set of comparable angles that are congruent. As a result, the converse of the parallel lines is true.

Prove:

If one of two parallel lines is perpendicular, the line is also perpendicular to the other.

Given:

 L3 ⊥ L1, L1 || L2, then prove L3 ⊥ L2

Proof

Here's how you prove the Perpendicular Transversal Theorem:

(1)  L3 ⊥ L1               // given

(2) m ∠ 1 = 90°           // definition of perpendicular lines

(3) L1 || L2                 // given

(4)   m ∠ 1 = m ∠ 2        // corresponding angles of two parallel lines

(5)   m ∠ 2 = 90°           //from (2) & (4) , using algebraic substitution

(6)   L3 ⊥ L2              // definition of perpendicular lines

Approach to this problem:

We'll prove this by calculating matching angles, just like practically every other geometry problem involving parallel lines. We'll use all of the facts provided in the problem statement, just like any other proof problem.

So, we know that L1 and L2 are parallel, so we utilize that knowledge to find the corresponding angles (1 and 2), which are equal. As L3 and L1 are perpendicular lines, we were also told that m1 = 90°. So, we have m1 = m2= 90° applying both facts, and we are done.

Perpendicular Transversal Theorem Corollary

Transversal perpendicular

Corollary 1: If one of the parallel lines is perpendicular to a transversal, it is also perpendicular to the other parallel line.

Given:

• Lines a and b are perpendicular to one another.
• Line c runs parallel to line a.
• The three lines are parallel. 

Frequently Asked Question by Students

Q.1. What is the parallel transversal theorem?

ANS.

When a transversal intersects two parallel lines, the two corresponding angles are equal.

From Fig.: ∠1=∠6, ∠4=∠8, ∠2= ∠5 and ∠3= ∠7

The converse of this axiom is also true: if two comparable angles are equal, the specified lines are parallel.

Theorem 1:

When two parallel lines intersect with a transversal, the pair of alternating interior angles are equal.

From Fig.: ∠4=∠5 and ∠3=∠6

Proof:

As ∠4=∠2 and ∠1=∠3 (Vertically Opposite Angles)

Also, ∠2=∠5 and ∠1=∠6 (Corresponding Angles)

⇒∠4=∠5 and ∠3=∠6

The converse of the preceding theorem is also true, which asserts that if two opposite interior angles are equal, the lines are parallel.

Theorem 2:

When a transversal intersects two parallel lines, the pair of internal angles on the same side of the transversal are supplementary.

∠3+ ∠5=180° and ∠4+∠6=180°

As ∠4=∠5 and ∠3=∠6 (Alternate interior angles)

∠3+ ∠4=180° and ∠5+∠6=180° (Linear pair axiom)

⇒∠3+ ∠5=180° and ∠4+∠6=180°

The converse of the aforementioned theorem holds true, stating that if a pair of co-interior angles are supplementary, the given lines are parallel.

Q.2. Are perpendicular lines always 90 degrees?

ANS. The quality of being perpendicular (perpendicularity) in elementary geometry is the relationship between two lines that meet at a right angle, 90°. When two lines intersect at a straight angle, the two lines are said to be perpendicular.

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