In today’s lesson, we’ll prove the Linear Pair Perpendicular Theorem: This theorem states that if two straight lines intersect at a point, and the linear pair of angles they form have an equal measure, then the two lines are perpendicular to each other.
Problem
If two straight lines intersect at a point and form a linear pair of equal angles, they are perpendicular.
in other words, given m∠1 = m∠2, prove that the lines L1 and L2 are perpendicular.
Strategy for proving the Linear Pair Perpendicular Theorem
To prove that lines are perpendicular, we need to find an angle that measures 90°. In the problem statement, we are given the fact that we have a linear pair of angles, and that they are equal. The hint here is “linear pair” – we know those measure 180°, and 180° is twice 90°.
Proof
(1) m∠1 = m∠2
(2) m∠1 + m∠2 = 180° // straight line measures 180°
(3) m∠1 + m∠1 = 180° // using (2) and performing algebraic substitution, replacing m∠2 with the equivalent m∠1
(4) 2 • m∠1 = 180° // Simplify, using multiplication
(5) m∠1 = 180°/2= 90° //algebraically solve for m∠1
(6) L1 ⊥ L2 //per the definition of perpendicular lines.
