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 1-4: Angles 7

Measuring Angles

Just as we can measure segments, we can also measure angles.

We use units called degrees to measure angles.

• A circle measures _____

• A (semi) half-circle measures _____

• A quarter-circle measures _____

• One degree is the angle measure of 1/360th of a circle.

?

?

?

360º

180º

90º

Lesson 1-4: Angles 8

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 9

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 10

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!

Lesson 1-4: Angles 11

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 12

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 13

Example

Draw your own diagram and answer this question: If is the angle bisector of PMY and mPML = 87,

then find: mPMY = _______ mLMY = _______

ML