Converse Perpendicular Transversal Theorem

Lines Perpendicular to a Transversal Theorem 3 4
Lines Perpendicular to a Transversal Theorem 3 4

In today’s lesson, we will learn a step-by-step proof of the Converse Perpendicular Transversal Theorem: If two lines are perpendicular to a 3rd line, then they are parallel to each other.

In the previous problem, we showed that if a transversal line is perpendicular to one of two parallel lines, it is also perpendicular to the other parallel line.

Now we will show the opposite – that if a transversal line is perpendicular to two lines, then those two lines are parallel.

Prove:

If two lines are both perpendicular to one other line, then they are parallel to each other.

Given:

L3 ⊥ L1, L3 ⊥ L2, prove L1 || L2

Proof

(1) L3 ⊥ L1 //given

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

(3) L3 ⊥ L2 //given

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

(5) m∠1 = m∠2 // transitive property of equality, both equal 90°

(6) ∠1 ≅ ∠2 //from (5) and definition of congruency

L1 || L2 //two lines are parallel if the corresponding angles formed by a transversal line are congruent

Strategy for this problem:

To show that two lines are parallel, we typically need to find two corresponding angles that are equal. The corresponding angles here are ∠1 ND ∠2, and using the facts given in the problem – that these are both right angles (since both L1 and L2 lines are perpendicular to L3), they are equal.

And that’s how we prove the Converse Perpendicular Transversal Theorem.

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