Download Perpendicular Lines

Perpendicular Lines
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
Bill Zahner
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Printed: November 15, 2015
AUTHORS
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
Bill Zahner
www.ck12.org
C HAPTER
Chapter 1. Perpendicular Lines
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Perpendicular Lines
Here you’ll learn what perpendicular lines are and how to apply some basic properties and theorems about such
lines.
What if you were given a pair of lines that intersect each other at a 90◦ 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.
Watch This
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/136581
CK-12 Perpendicular Lines
Watch the portions of this video dealing with perpendicular lines.
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/1341
James Sousa: Perpendicular Lines
Then watch this video.
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/1344
James Sousa: Perpendicular Line Postulate
Guidance
Two lines are perpendicular when they intersect to form a 90◦ angle. Below, l ⊥ AB.
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In the definition of perpendicular the word “line” is used. However, line segments, rays and planes can also be
perpendicular. The image below shows two parallel planes, with a third blue plane that is perpendicular to both of
them.
Basic Facts about Perpendicular Lines
Theorem #1: If l||m and n ⊥ l, then n ⊥ m.
Theorem #2: If l ⊥ n and n ⊥ m, then l||m.
Postulate: For any line and a point not on the line, there is one line perpendicular to this line passing through the
point. There are infinitely many lines that pass through A, but only one that is perpendicular to l.
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Chapter 1. Perpendicular Lines
Example A
Which of the following is the best example of perpendicular lines?
1.
2.
3.
4.
Latitude on a Globe
Opposite Sides of a Picture Frame
Fence Posts
Adjacent Sides of a Picture Frame
The best example would be adjacent sides of a picture frame. Remember that adjacent means next to and sharing a
vertex. The adjacent sides of a picture frame meet at a 90◦ angle and so these sides are perpendicular.
Example B
←
→ −→
Is SO ⊥ GD?
OGD ∼
= 6 SGD and the angles form a linear pair. This means both angles are 90◦ , so the lines are perpendicular.
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Example C
Write a 2-column proof to prove Theorem #1. Note: You need to understand corresponding angles in order to
understand this proof. If you have not yet learned corresponding angles, be sure to check out that concept first, or
skip this example for now.
Given: l||m, l ⊥ n
Prove: n ⊥ m
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TABLE 1.1:
Statement
1. l||m, l ⊥ n
2. 6 1, 6 2, 6 3, and 6 4 are right angles
3. m6 1 = 90◦
4. m6 1 = m6 5
5. m6 5 = 90◦
6. m6 6 = m6 7 = 90◦
7. m6 8 = 90◦
8. 6 5, 6 6, 6 7, and 6 8 are right angles
9. n ⊥ m
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
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/136582
CK-12 Perpendicular Lines
–>
Guided Practice
1. Find m6 CTA.
2. Determine the measure of 6 1.
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Chapter 1. Perpendicular Lines
3. Find m6 1.
Answers:
1. These two angles form a linear pair and 6 STC is a right angle.
m6 STC = 90◦
m6 CTA is 180◦ − 90◦ = 90◦
2. We know that both parallel lines are perpendicular to the transversal.
m6 1 = 90◦ .
3. The two adjacent angles add up to 90◦ , so l ⊥ m.
m6 1 = 90◦
because it is a vertical angle to the pair of adjacent angles and vertical angles are congruent.
Explore More
Use the figure below to answer questions 1-2. The two pentagons are parallel and all of the rectangular sides are
perpendicular to both of them.
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1. List a pair of perpendicular lines.
2. For AB, how many perpendicular lines would pass through point V ? Name this/these line(s).
Use the picture below for question 3.
3. If t ⊥ l, is t ⊥ m? Why or why not?
Find the measure of 6 1 for each problem below.
4.
5.
6.
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Chapter 1. Perpendicular Lines
7.
8.
9.
10.
11.
12.
In questions 13-16, determine if l ⊥ m.
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13.
14.
15.
16.
Fill in the blanks in the proof below.
17. Given: l ⊥ m, l ⊥ nProve: m||n
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Chapter 1. Perpendicular Lines
TABLE 1.2:
Statement
1.
2. 6 1 and 6 2 are right angles
3.
4.
5. m||n
Reason
1.
2.
3. Definition of right angles
4. Transitive PoE
5.
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 3.2.
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