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Lesson 1-4: Angles 1 Lesson 1-4 Angles
Lesson 1-4: Angles 1
Lesson 1-4
Angles
Lesson 1-4: Angles 2
Angle and Points
An Angle is a figure formed by two rays with a common endpoint, called the vertex.
vertex
ray
ray Angles can have points in the interior, in the exterior or on the
angle.
Points A, B and C are on the angle. D is in the interior and E is in the exterior. B is the vertex.
A
BC
DE
Lesson 1-4: Angles 3
Naming an angle: (1) Using 3 points (2) Using 1 point (3) Using a number – next slide
ABC or CBA
Using 3 points: vertex must be the middle letter
This angle can be named as
Using 1 point: using only vertex letter
* Use this method is permitted when the vertex point is the vertex of one and only one angle.
Since B is the vertex of only this angle, this can also be called .
A
B C
B
Lesson 1-4: Angles 4
Naming an Angle - continued
Using a number: A number (without a degree symbol) may be used as the label or name of the angle. This number is placed in the interior of the angle near its vertex. The angle to the left can be named
as .2
* The “1 letter” name is unacceptable when …more than one angle has the same vertex point. In this case, use the three letter name or a number if it is present.
2
A
B C
Lesson 1-4: Angles 5
Example
K
32
K
L
M
P
Therefore, there is NO in this diagram.There is , ,LKM PKM and LKP
2 3 5!!!There is also and but there is no
K is the vertex of more than one angle.
Lesson 1-4: Angles 6
4 Types of Angles
Acute Angle: an angle whose measure is less than 90.
Right Angle: an angle whose measure is exactly 90 .
Obtuse Angle: an angle whose measure is between 90 and 180.
Straight Angle: an angle that is exactly 180 .
Lesson 1-5: Pairs of Angles 7
Complementary AnglesA pair of angles whose sum is 90˚Definition:
Examples:
Adjacent Angles( a common side )
21
Q
AB
C 1
2
Q
R
AB
F
G
Non-Adjacent Angles
m1 = 40°m2 = 50°
Lesson 1-5: Pairs of Angles 8
Supplementary AnglesA pair of angles whose sum is 180˚Definition:
Examples:
Adjacent supplementary angles are also called “Linear Pair.”
Non-Adjacent Angles
2 1
A Q
B
C
1
2
A QR
BF
Gm2 = 140°
m1 = 40°
Lesson 1-5: Pairs of Angles 9
Vertical AnglesA pair of angles whose sides form opposite rays.Definition:
4
3
2
1A
Q
D
B
C
Examples:
2 and 4
1 and 3
Vertical angles are non-adjacent angles formed by intersecting lines.
Lesson 1-5: Pairs of Angles 10
Theorem: Vertical Angles are =
The diagramGiven:
4
3
2
1A
Q
D
B
C
Prove:
~
1 3
Statements Reasons
m2 + m3 = 180°m1 + m2 = 180°1.
m1 + m2 = m2 + m32.
m1 = m33.
m1 m34.
1. Definition: Linear Pair
2. Property: Substitution
3. Property: Subtraction
4. Definition: Congruence
Lesson 1-5: Pairs of Angles 11
Adjacent Angles
A pair of angles with a shared vertex and common side but do not have overlapping interiors.
1 and 2 are adjacent. 3 and 4 are not. 1 and ADC are not adjacent.
Adjacent Angles( a common side ) Non-Adjacent Angles
22°
36°
21
D
B
C
A4
3
Definition:
Examples:
Lesson 1-4: Angles 12
Adding Angles
When you want to add angles, use the notation m1, meaning the measure of 1.
If you add m1 + m2, what is your result?
m1 + m2 = 58.
22°
36°
21
D
B
C
A
Therefore, mADC = 58.
m1 + m2 = mADC also.
Lesson 1-4: Angles 13
Angle Addition Postulate
R
M K
W
The sum of the two smaller angles will always equal the measure of the larger angle.
Complete:
m ____ + m ____ = m _____MRK KRW MRW
Postulate:
Lesson 1-4: Angles 14
Angle Addition Postulate
R
M K
W
The sum of the two smaller angles will always equal the measure of the larger angle.
Complete:
m ____ + m ____ = m _____MRK KRW MRW
Postulate:
Lesson 1-4: Angles 15
Angle Bisector
An angle bisector is a ray in the interior of an angle that splits the angle into two congruent angles.
UK
j41°
41°
64
U
K53
Example: Since 4 6, is an angle bisector.
Lesson 1-4: Angles 16
3 5.
Congruent Angles
53
Definition: If two angles have the same measure, then they are congruent.
Congruent angles are marked with the same number of “arcs”.
The symbol for congruence is
Example:
Lesson 1-4: Angles 17
Example
Draw your own diagram and answer this question: If is the angle bisector of PMY and mPML = 87,
then find: mPMY = _______ mLMY = _______
ML
Lesson 1-5: Pairs of Angles 18
What’s “Important” in Geometry?
360˚ 180˚ 90˚
4 things to always look for !
. . . and CongruenceMost of the rules (theorems)and vocabulary of Geometryare based on these 4 things.
Lesson 1-5: Pairs of Angles 19
Example: If m4 = 67º, find the measures of all other angles.
3 4 180m m
3 67 180m
3 180 67 113m
4
3
2
1
67º
Step 1: Mark the figure with given info.
Step 2: Write an equation.
3 1 , . 3 1 117 Because and are vertical angles they are equal m m
4 2 , . 4 2 67 Because and arevertical angles they are equal m m
Lesson 1-5: Pairs of Angles 20
Example: If m1 = 23 º and m2 = 32 º, find the measures of all other angles.
4 23 ( 1 & 4 .)
5 32 ( 2 & 5 .)
m are vertical angles
m are vertical angles
6 5
4 3 2
1
Answers:
1 2 3 180
23 32 3 180
3 180 55 125
3 6 125
3 & 6 .
m m m
m
m
m m
are vertical angles
Lesson 1-5: Pairs of Angles 21
Example: If m1 = 44º, m7 = 65º find the measures of all other angles.
3 90m
1 4 44m m
4 5 90
44 5 90
5 46
m m
m
m
7
6 5 4
3
2 1
Answers:
6 7 90
6 65 90
6 25
m m
m
m
Lesson 1-5: Pairs of Angles 22
Algebra and Geometry
( ) = ( )( ) + ( ) = ( )( ) + ( ) = 90˚( ) + ( ) = 180˚
Common Algebraic Equations used in Geometry:
If the problem you’re working on has a variable (x),then consider using one of these equations.
