•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.