3.2. Perpendicular Lines http://www.ck12.org
Perpendicular Transversals
Recall that when two lines intersect, four angles are created. If the two lines are perpendicular, then all four angles
are right angles, even though only one needs to be marked with the square. Therefore, all four angles are 90◦.
When a parallel line is added, then there are eight angles formed. Ifl||mandn⊥l, isn⊥m? Let’s prove it here.
Given:l||m,l⊥n
Prove:n⊥m
TABLE3.1:
Statement Reason
1.l||m,l⊥n Given2.^61 ,^62 ,^6 3, and^6 4 are right angles Definition of perpendicular lines
3.m^61 = 90 ◦ Definition of a right angle
4.m^61 =m^65 Corresponding Angles Postulate
5.m^65 = 90 ◦ Transitive PoE
6.m^66 =m^67 = 90 ◦ Congruent Linear Pairs
7.m^68 = 90 ◦ Vertical Angles Theorem
8.^65 ,^66 ,^6 7, and^6 8 are right angles Definition of right angle
9.n⊥m Definition of perpendicular lines
Theorem #1: If two lines are parallel and a third line is perpendicular to one of the parallel lines, it is also
perpendicular to the other parallel line.
Or, ifl||mandl⊥n, thenn⊥m.
Theorem #2:If two lines are perpendicular to the same line, they are parallel to each other.
Or, ifl⊥nandn⊥m, thenl||m. You will prove this theorem in the review questions.
From these two theorems, we can now assume that any angle formed by two parallel lines and a perpendicular
transversal will always be 90◦.
Example A
Findm^6 CT A.