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3.4 Parallel and Perpendicular Lines
Objectives : G.CO.9 : Prove geometric theorems about lines and angles.
G.CO.12 : Make formal geometric constructions with a variety of tools and methods.
For the Board : You will be able to prove and apply theorems about perpendicular lines.
Bell Work 3.4 :
Solve each inequality.
- x โ 5 < 8 2. 3x + 1 > x
Solve each equation.
- 5y = 90 4. 5x + 15 = 90
Solve the system of equations.
- 6y = 90 and 8y โ 3x = 90
Anticipatory Set :
Perpendicular lines are lines that intersect to form right angles.
The shortest distance from a point to a line is along
the perpendicular to the line from the point.
So the distance from a point to a line is defined as
the length of the perpendicular segment from
the point to the line.
Open the book to page 172 and read example 1.
Practice 1: a. Name the shortest segment from point A to BC.
Write and solve an inequality for x.
AP x โ 8 > 12 or x > 20
b. Write and solve an inequality for x.
7x + 4 < 10x โ 2
-3x < -
x > 2
Instruction :
Congruent Linear Pairs โ Perpendicular Lines
If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular.
Given: <1 ๏<2 and <1 and <2 form a linear pair
Prove: l | m
Proof: <1 and <2 must be right angles, therefore
l | m by the definition of perpendicular.
P
B
A
C
x - 8
m
l
10x - 2
7x + 4
Parallel Transitive Theorem
If two lines are parallel to the same line, then they are parallel.
Given: l || m and m || n
Prove: l || n
Perpendicular Transversal Theorem
If a transversal is perpendicular to one of two parallel lines,
then it is perpendicular to the other.
Given: l | h , h || k
Prove: l | k
Two Perpendiculars Theorem
In a plane, if two lines are perpendicular to the same line,
then they are parallel to each other.
Given: l | h , l | k
Prove: h || k
Practice:
a. Given <3 ๏<4 what can be concluded about t and l?
What postulate or theorem applies?
t | l, Congruent Linear Pairs Theorem
b. Given l||n and t | n, what can be concluded about lines t and l?
What postulate or theorem applies?
t | n, Perpendicular Transversal Theorem
c. Given l||m and l||n, what can be concluded about lines m and n?
What postulate or theorem applies?
m||n, Parallel Transitive Theorem
d. Given <6 ๏<8 what can be concluded about t and m? What postulate or theorem applies?
t | m, Congruent Linear Pairs Theorem
e. Given t | l and t | n, what can be concluded about l and n? What postulate or theorem applies?
l||n, Two Perpendiculars Theorem
f. Given < ๏
<5 and < ๏
<10, what can be concluded about lines l and n? What postulate or
theorem applies?
Since < ๏
<5, l||m by the Corresp. <โs Post. Since < ๏
<10, m||n by the Alt. Int. <โs Th.
Therefore, l||n by the Parallel Transitive Theorem.
g. Given < ๏
<6 and <1 is a right angle, what can be concluded about t and m? What postulate or
theorem applies.
Since < ๏
<6, l||m by the Alt. Int. <โs Th. Since <1 is a rt. <, t | l by the defn. of |.
Therefore, t | m by the Perpendicular Transversal Theorem.
h
k
l
1
2
3
t
n
m
l
1 2
3 4
5 6
7
910
8
111 2
n
m
l
t
Read example 2 on page 173 then with your partner complete practice 2.
Practice 2: Write a two-column proof.
Given: r||s, < ๏
Prove: r | t
Proof:
**1. r||s 1. Given
- <3** ๏ <2 2. Corresponding Angles Theorem
<2 3. Given
<3 4. Transitive Property of Congruence
5. r | t 5. Congruent Linear Pair of Angles form Perpendicular Lines
Assessment :
Question Student Pairs.
Independent Practice :
Text: pg. 175-178 prob. 2, 3, 6, 7, 10 โ 15.
For a Grade :
Handout 3.
r
s
t
