Angles
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Angles. 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. B is the vertex. E. D. Points A, B and C are on the angle. B. C.

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Angles

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Angles

Angles


Angle and points

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

B is the vertex.

E

D

Points A, B and C are on the angle.

B

C

D is in the interior

E is in the exterior.


Naming an a ngle

Naming an Angle

Using 3 points:

Vertex must be the middle letter

This angle can be named as

Using 1 point:

Using only vertex letter

Using a number:

A

Use the notation m2, meaning the measure of 2.

B

C


Example

Example

Name all the angles in the diagram below

K is the vertex of more than one angle.

Therefore, there is NO in this diagram.


Example1

Example

Name the three angles in the diagram.


4 types of angles

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 .


Angle addition postulate same idea as the segment addition postulate

Angle Addition PostulateSame idea as the segment addition postulate

Postulate:

The sum of the two smaller angles will always equal the measure of the larger angle.

Complete:

m  ____ + m ____ = m  _____

MRK

KRW

MRW


Example2

Example

Fill in the blanks.

m < ______ + m < ______ = m < _______


Adding angles

Adding Angles

If you add m1 + m2, what is your result?

m1 + m2 = 58.

Also…

m1 + m2 = mADC

Therefore, mADC = 58.


Example3

Example

K is interior to MRW, m  MRK = (3x), m KRW = (x + 6) and mMRW = 90º. Find mMRK.

First, draw it!

3x + x + 6 = 90

4x + 6 = 90

– 6 = –6

4x = 84

x = 21

3x

x+6

Are we done?

mMRK = 3x = 3•21 = 63º


Example4

Example

Given that m< LKN = 145, find m < LKM and m < MKN


Angles

Example

Given that < KLM is a straight angle, find m < KLN and m < NLM


Example5

Example

Given m < EFG is a right angle, find m < EFH and m < HFG


Angle bisector

Angle Bisector

An angle bisector is a ray in the interior of an angle that splits the angle into two congruent angles.

5

3


Congruent angles

Congruent Angles

Definition:

If two angles have the same measure, then they are congruent.

Congruent angles are marked with the same number of “arcs”.

3

5

3   5.


Example6

Example:

is an angle bisector

J

T

Which two angles are congruent?

<JUK and < KUT or < 4 and < 6


Example7

Example:

Given bisects < XYZ and m < XYW = . Find m < XYZ


Example8

Example:

Given bisects < ABC. Find m < ABC


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