unit: tools of geometry lesson: 1.2a – angles
DESCRIPTION
Angle and Points ray vertex ray 4/27/2017 Angle and Points An Angle is a figure formed by two rays with a common endpoint, called the vertex. ray vertex ray Angles can have points in the interior, in the exterior or on the angle. A E D B C Points A, B and C are on the angle. D is in the interior and E is in the exterior. B is the vertex.TRANSCRIPT
•9/17/15•CC Geometry•UNIT: 1.1 - Tools of Geometry •LESSON: 1.2a – Angles•MAIN IDEA: Students will be able to use information to determine the measure of angles and explain relationships between angles.
•HOMEWORK: Textbook • Page 41 #’s 9-10• Page 51 #’s 19-26• Take home quiz due Monday 9/21
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
Naming an angle: (1) Using 3 points (2) Using 1 point (3) Using a number – next slide
ABC or CBA
B
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
Naming an Angle - continued
2
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 .
* 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
Example
K, ,LKM PKM and LKP
32
K
LM
P
Therefore, there is NO in this diagram.There is
2 3 5!!!There is also and but there is no
K is the vertex of more than one angle.
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 .
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°
21D
B
C
A
Therefore, mADC = 58.
m1 + m2 = mADC also.
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:
Example: Angle Addition
R
M K
W
3x + x + 6 = 90 4x + 6 = 90 – 6 = –64x = 84x = 21
K is interior to MRW, m MRK = (3x), m KRW = (x + 6) and mMRW = 90º. Find mMRK.
3xx+6 Are we done?
mMRK = 3x = 3•21 = 63º
First, draw it!
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.
Congruent Angles
3 5.53
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:
Example
• Draw your own diagram and answer this question:• If is the angle bisector of PMY and mPML = 87, then find:
• mPMY = _______• mLMY = _______
ML
Complementary AnglesTwo angles are called complementary angles if the sum of their degree measurements equals 90 degrees.
Example: These two angles are complementary. Their sum is 90˚.
58° + 32° = 90°
These two angles can be "pasted" together to form a right angle! Adjacent complementary angles.
Complementary Angles
Supplementary AnglesTwo angles are called supplementary angles if the sumof their degree measurements equals 180 degrees.
Example: These two angles are supplementary. The sum of their measures is 180˚
139° +41° = 180 °
Two angles that share a vertex and a side to form a line.
Two angles that can be "pasted" together to form a straight line!
Linear Pair of AnglesSpecial Supplementary Angles
Vertical AnglesOpposite angles formed by intersecting lines .
For any two lines that meet, such as in the diagram below, angle AEB and angle DEC are called vertical angles. Angle BEC and angle AED are also vertical angles. Vertical angles are congruent - have the same degree measurement.
Review
State whether the following are acute, right, or obtuse.
1.
2.
3.
4.
5.
??
acute obtuseright
obtuseacute
Complementary and Supplementary
1. Two angles are complementary. One measures 65 degrees.
2. Two angles are supplementary. One measures 140 degrees.
Find the missing angle.
Answer : 25
Answer : 40
Complementary and Supplementary
Find the missing angle. You do not have a protractor.Use the clues in the pictures.1. 2.x
55 165x
x = 90° – 55° x = 180° – 165°
x = 35 ° x = 15°
Vertical Angles
Find the missing angle.
58x x = 58
More drawings
20
C
J
D
E F
G
H
70
9070
20
90 Box in the corner indicates a right angle.
Final Drawing
52
B
A
F E
D
C
60
G
68
686052
Find the measure of each missing angle
Exit Ticket
Explain why angle 1 is congruent to angle 3.