3.2: Perpendicular Lines
What if you were given a pair of lines that intersect each other at a right angle? What terminology would you use to describe such lines? After completing this Concept, you will be able to define perpendicular lines. You'll also be able to apply the properties associated with such lines to solve for unknown angles
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CK12 Foundation: Chapter3PerpendicularLinesA
Watch the portions of this video dealing with perpendicular lines.
James Sousa: Perpendicular Lines
James Sousa: Perpendicular Line Postulate
Guidance
Two lines are
perpendicular
if they meet at a
Investigation: Perpendicular Line Construction; through a Point NOT on the Line
1. Draw a horizontal line and a point above that line.
Label the line
2. Take the compass and put the pointer on
3. Move the pointer to one of the arc intersections. Widen the compass a little and draw an arc below the line. Repeat this on the other side so that the two arc marks intersect.
4. Take your straightedge and draw a line from point
Notice that this is a different construction from a perpendicular bisector.
To see a demonstration of this construction, go to: http://www.mathsisfun.com/geometry/constructperpnotline.html
Investigation: Perpendicular Line Construction; through a Point on the Line
1. Draw a horizontal line and a point on that line.
Label the line
2. Take the compass and put the pointer on
3. Move the pointer to one of the arc intersections. Widen the compass a little and draw an arc above or below the line. Repeat this on the other side so that the two arc marks intersect.
4. Take your straightedge and draw a line from point
Notice that this is a different construction from a perpendicular bisector.
To see a demonstration of this construction, go to: http://www.mathsisfun.com/geometry/constructperponline.html
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
When a parallel line is added, then there are eight angles formed. If
Given
:
Prove
:
Statement  Reason 

1.

Given 
2.

Definition of perpendicular lines 
3.

Definition of a right angle 
4.

Corresponding Angles Postulate 
5.

Transitive PoE 
6.

Congruent Linear Pairs 
7.

Vertical Angles Theorem 
8.

Definition of right angle 
9.

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, if
Theorem #2: If two lines are perpendicular to the same line, they are parallel to each other.
Or, if
From these two theorems, we can now assume that any angle formed by two parallel lines and a perpendicular transversal will always be
Example A
Find
First, these two angles form a linear pair. Second, from the marking, we know that
Example B
Determine the measure of
From Theorem #1, we know that the lower parallel line is also perpendicular to the transversal. Therefore,
Example C
Find
The two adjacent angles add up to
Watch this video for help with the Examples above.
CK12 Foundation: Chapter3PerpendicularLinesB
Vocabulary
Two lines are
perpendicular
if they meet at a
Guided Practice
1. Is
2. Find the value of
3. Find the value of
Answers:
1. If the two adjacent angles add up to
2. The two angles together make a right angle. Set up an equation and solve for
3. The two angles together make a right angle. Set up an equation and solve for
Practice
Find the measure of
In questions 1013, determine if
Find the value of
Image Attributions
Description
Learning Objectives
Here you'll learn about perpendicular lines, including their properties and how to construct them.