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.
If two lines are both perpendicular to one other line, then they are parallel to each other.
L3 ⊥ L1, L3 ⊥ L2, prove L1 || L2
(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.