Download Chapter 1 Points, Lines, Planes, and Angles page 1

Lesson 2-5
Perpendicular Lines
(page 56)
Essential Question
Can you justify the conclusion
of a conditional statement?
Perpendicular Lines (⊥-lines):
… two lines that intersect to
form right angles.
Example:
m
n
If m ⊥n,
∠4
then _____,
∠1 _____,
∠2 _____,
∠3 and _____
are right angles.
Example:
1
2
4
3
m
n
How many angles must be right angles in
order for the lines to be perpendicular?
1
_____
Theorem 2-4
⊥-lines ⇒ ≅ adj∠’s
If two lines are perpendicular,
then they form congruent adjacent angles.
Given:
l⊥n
Prove:
∠1, ∠2, ∠3, & ∠4
are congruent angles.
l
2
1
3
4
n
l
Given: l ⊥ n
Prove: ∠1, ∠2, ∠3, & ∠4
are congruent angles.
Statements
2
1
3
4
n
Reasons
1. ___________________ ___________________
See page
57
2. ___________________
___________________
Classroom Exercises
3. ___________________ ___________________
#1 for HELP!
4. ___________________ ___________________
l
Given: l ⊥ n
Prove: ∠1, ∠2, ∠3, & ∠4
are congruent angles.
Statements
2
1
3
4
n
Reasons
1. ___________________
___________________
l⊥n
Given
2. ∠1,
___________________
∠2, ∠3, & ∠4 Rt.∠’s ___________________
Def. of ⊥-lines
3. ∠1,
___________________
∠2, ∠3, & ∠4 90º∠’s
∠2, ∠3, & ∠4 ≅ ∠’s
4. ∠1,
______________________
___________________
Def. of Rt.∠’s
___________________
Def. of ≅ ∠’s
Theorem 2-5
2 lines form ≅ adj∠’s
⇒⊥-lines
If two lines form congruent adjacent angles,
then they are perpendicular .
Given: ∠1 ≅ ∠2
Prove: l ⊥ n
l
2
1
n
Note: This is the converse of Theorem 2-4.
l
Given: ∠1 ≅ ∠2
Prove: l ⊥ n
Statements
2
1
n
Reasons
1. ___________________ ___________________
See page
58
2. ___________________
___________________
Written
3. ___________________
___________________
Exercises
___________________
#2 for HELP!
4. ___________________ ___________________
5. ______________________
___________________
l
Given: ∠1 ≅ ∠2
Prove: l ⊥ n
Statements
2
1
n
Reasons
1.∠1
___________________
Given
≅ ∠2, or m∠1 = m∠2 ___________________
2. ___________________
___________________
Angle Add. Post.
m∠1 + m∠2 = 180º
3. ___________________
___________________
m∠2 + m∠2 = 180º
Substitution Prop.
___________________
( 2 m∠2 = 180º )
(Distributive
Prop.)
Division Prop.
4. ___________________
___________________
m∠2 = 90º
5. ______________________
___________________
l⊥n
Def. of ⊥-lines
Theorem 2-6
Ext S 2 adj A∠’s ⊥
⇒ comp ∠’s
If the exterior sides of two adjacent
acute angles are perpendicular ,
then the angles are complementary.
Given:
Prove:
OA ^ OC
A •
∠AOB and ∠BOC
B
are comp. ∠‘s
O
•
•
C
Given:
OA ^ OC
Prove:
∠AOB and ∠BOC
A •
B
are comp. ∠‘s
•
O
Statements
1.
2.
3.
4.
•
C
Reasons
___________________ ___________________
See page
59
___________________
___________________
Written
___________________
___________________
Exercises
___________________ ___________________
#13 for HELP!
5. ______________________
___________________
Given:
OA ^ OC
Prove:
∠AOB and ∠BOC
A •
B
are comp. ∠‘s
•
O
Statements
1.
2.
3.
4.
___________________
OA ^ OC
___________________
m∠AOC = 90º
___________________
m∠AOB + m∠BOC = m∠AOC
___________________
m∠AOB + m∠BOC = 90º
5. ∠AOB
______________________
& ∠BOC are comp. ∠‘s
•
C
Reasons
___________________
Given
___________________
Def. of ⊥-lines
___________________
Angle Add. Post.
___________________
Substitution Prop.
Def. of comp. ∠‘s
___________________
Example:
Name the definition or theorem that justifies
each statement about the diagram.
B
1
2
A
3
5
4
6
C
8
7
D
(a)
If AB^ BC, then ÐABC is a right angle.
Def. of ⊥-lines
_________________________________
Example:
Name the definition or theorem that justifies
each statement about the diagram.
B
1
2
A
3
5
4
6
C
8
7
D
(b)
If BD ^ AC, then Ð3@ Ð4.
⊥-lines ⇒ ≅ adj∠’s
_________________________________
Example:
Name the definition or theorem that justifies
each statement about the diagram.
B
1
2
A
3
5
4
6
C
8
7
D
(c)
If DC ^ DA, then Ð7 and Ð8 are complementary.
Ext _________________________________
S 2 adj A∠’s ⊥ ⇒ comp ∠’s
Example:
Name the definition or theorem that justifies
each statement about the diagram.
B
1
2
A
3
5
4
6
C
8
7
D
(d)
If ∠7 & ∠8 are complementary,
then m∠7 + m∠8 = 90º.
Def. of comp ∠’s
_________________________________
Example:
Name the definition or theorem that justifies
each statement about the diagram.
B
1
2
A
3
5
4
6
C
8
7
D
(e)
If Ð4 @ Ð6, then AC ^ BD.
2 _________________________________
lines form ≅ adj∠’s ⇒⊥-lines
Example:
Name the definition or theorem that justifies
each statement about the diagram.
B
1
2
3
5
A
4
6
C
8
7
D
(f)
∠4 ≅ ∠5.
Vertical ∠’s R ≅
_________________________________
Example:
Name the definition or theorem that justifies
each statement about the diagram.
B
1
2
A
4
6
3
5
C
8
7
D
(g)
If ∠ABC is a right angle,
then m∠ABC = 90º.
Def. of Rt. ∠
_________________________________
If ZW ⊥ ZY, m∠1 = 5x, and m∠2 = 2x - 1,
find the value of x.
Example.
13
x = ______
W•
V
1
Z
2
•
mÐ1 + mÐ2 = 90º
5x + 2x -1= 90
7x = 91
x
=13
mÐ2 = 2x -1
•
Y
mÐ1= 5x
mÐ1= 5(13) mÐ2 = 2(13)-1 65º+25º
mÐ1= 65º mÐ2 = 26-1
= 90º
mÐ2 = 25º
Assignment
Written Exercises on pages 58 & 59
RECOMMENDED: 15 & 17
REQUIRED: 3 to 11 odd numbers,
19 to 25 odd numbers
Prepare for a Quiz on
Lessons 2-2 to 2-5: Justifications
Can you justify the conclusion
of a conditional statement?