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Lesson 2-4
Special Pairs of Angles
(page 50)
Essential Question
Can you justify the conclusion
of a conditional statement?
Complementary Angles:
… two angles whose measures
have the sum 90 degrees.
A
60º
Example:
mÐA + mÐB = 90º
30º
C
B
\ ÐA & ÐB are complementary angles.
Supplementary Angles:
… two angles whose measures
have the sum 180 degrees.
Example:
1
2
mÐ1 + mÐ2 = 180º
\ Ð1 & Ð2 are supplementary angles.
In the diagram, ∠CXD is a right angle.
Example #1.
B•
A
•
E
X
•
C
•
•D
Name another right angle.
ÐAXD
_________________________________
In the diagram, ∠CXD is a right angle.
Example #1.
B•
A
•
E
X
•
C
•
•D
Name complementary angles.
ÐAXE & ÐEXD
_________________________________
In the diagram, ∠CXD is a right angle.
Example #1.
B•
A
•
E
X
•
C
•
•D
Name 2 congruent supplementary angles.
ÐAXD & ÐCXD
_________________________________
In the diagram, ∠CXD is a right angle.
Example #1.
B•
A
•
E
X
•
C
•
•D
Name 2 noncongruent supplementary angles.
ÐAXE & ÐEXC ; ÐAXE & ÐAXB ;
ÐAXB & ÐBXC ; ÐCXE & ÐCXB …
________________________________________________
Example # 2.
∠C and ∠ D are complementary,
m∠C = 3x - 5 and m∠D = x + 15.
Find the value of x, m∠C, and m∠D.
mÐC + mÐD = 90º
20
3x -5 + x +15 = 90
55º
4x +10 = 90 = 90º
4x = 80
35º
x
=
20
mÐC = 3x -5
mÐD = x +15
mÐC = 3(20)- 5
mÐD = 20 +15
mÐC = 55º
mÐD = 35º
x = ______
m∠C = _____
m∠D = _____
Example # 3.
∠P and ∠ K are supplementary,
m∠P = 4x + 10 and m∠K = x + 20.
Find the value of x, m∠P, and m∠K.
mÐP + mÐK = 180º
30
4x +10+ x + 20 =180
130º
5x + 30 =180 = 180º
5x =150
50º
x
=
30
mÐP = 4x +10
mÐK = x + 20
mÐP = 4(30)+10
mÐK = 30+ 20
mÐP =130º
mÐK = 50º
x = ______
m∠P = _____
m∠K = _____
Example #4. A supplement of an angle is seven times a
complement of the angle. Find the measures
of the angle, its complement, and its supplement.
measure of angle
x 75º
90 - x 15º
180 - x 105º
= __________ = ________
measure of complement
= __________ = ________
measure of supplement
= __________ = ________
180 - x = 7(90 - x)
180 - x = 630 - 7x
6x = 450
x = 75
Example #5. A complement of an angle is 15 less than twice
the measure of the angle. Find the measures
of the angle, its complement, and its supplement.
measure of angle
x 35º
90 - x 55º
180 - x 145º
= __________ = ________
measure of complement
= __________ = ________
measure of supplement
= __________ = ________
90 - x = 2x - 15
105 = 3x
35 = x
x = 35
Vertical Angles:
… two angles whose sides form
two pairs of opposite rays.
Example:
Ð1 & Ð3
_______________
4
and
1
2
3
Ð2 & Ð4
_______________
Theorem 2-3
Vert. ∠’s R ≅
Vertical angles are congruent.
Given:
∠1 and ∠2 are vertical angles
Prove:
∠1 ≅ ∠2
1
3
2
Given: ∠1 and ∠2 are vertical angles
Prove:
∠1 ≅ ∠2
Statements
1
3
2
Reasons
1. ∠1
___________________
and ∠2 are vertical angles ___________________
Given
2. ___________________
___________________
Angle Add. Post.
m∠1 + m∠3 = 180º
___________________
m∠2 + m∠3 = 180º
3. m∠1
___________________
___________________
Substitution Prop.
+ m∠3 = m∠2 + m∠3
4. ______________________
___________________
Reflexive Prop.
m∠3 = m∠3
5. ___________________
___________________
Subtraction Prop.
m∠1 = m∠2
6. ___________________
___________________
∠1 ≅ ∠2
Def. of ≅ ∠’s
Example #4.
Find the value of “x”.
35º
(2x-5)º
20
x = _______
2x - 5 = 35
2x = 40
x = 20
Example #5.
Find the value of “x”.
xº
35º
xº
55
x = _______
x + 35 = 90
x = 55
Example #6.
Find the value of “x”.
4xº
(x+30)º
10
x = _______
4x = x + 30
3x = 30
x =10
Example #7.
Find the values of “x” and “y”.
(2x+30)º
4yº
2x + 30 =120
2x = 90
x = 45
(x+y)º
120º
45
15
y = _______
x = _______
4y +120 =180
4y = 60
y = 15
Example #8.
Find the measure of each angle.
C•
•
A
90º-60º • D
= 30º
60º
90º
B
60º
•
E
30º
•G
•F
Assignment
Written Exercises on pages 52 to 54
RECOMMENDED: 1 to 17 odd numbers
REQUIRED: 19 to 31 odd numbers, 32 to 35
Assignment
Worksheet on Lesson 2-4
Prepare for a Quiz on
Lesson 2-4: Special Pairs of Angles
Can you justify the conclusion
of a conditional statement?