CABT Math 8 - Angles and Angle Measurements

The session shall begin shortly…

LET’S START

!

An angle consists of two different rays that have the same initial point. The rays are sides of the angles. The initial point is called the vertex.

Angles

vertex

sidesA

B

C

Angles & Angle Measurements

Naming Angles

A

B

C

Notation: We denote an angle with three points and “” symbol.

Naming:

Use the three points, with the vertex always in the middle. The angle can also be named with just the vertex point as long as it is a “stand alone angle”.

The angle shown can be named as:

BAC, CAB, A


1

Use the diagram below to answer the following questions.

a. Name the vertex. R

b. Name the sides of the angle.

c. Name the angle three different ways. SRT, TRS, R

R

S

TRS and RT

Check your understanding

Measure of an Angle

A

B

C

The MEASURE of an angle is the amount of rotation a side makes with respect to the other side. The unit is DEGREES ()

Notation: We put the letter “m” in front of the angle name to denote its measure.

The measure of the angle shown can be written as:

mBAC, mCAB, mA


Example: Here are some examples of angles and their degree measurements.

Measure of an Angle


ProtractorA device used to measure angles.

Measure of an Angle


Measure of an Angle


A

B

C

What is the measure of this angle?

ANSWER: 50

Measure of an Angle


A

B

C


ANSWER: 110

Measure of an Angle


A

B

C


ANSWER: 90

Measure of an Angle


A

B

C


ANSWER: 70

Reading a Protractor

Date: 01/13/2014 at 15:51:52 From: Donya InaSubject: Reading a Protractor

Anak, alam kong may protractor ka at marunong kang mag-measure ng angle. At least may pakinabang ‘yan kesa kung anu-ano ang ipino-post mo sa FB. Pero tanong ko lang: which side of a protractor scale do you use?

Paki-explain. Lab U!


Measure of an Angle

Date: 01/13/2014 at 15:52:53 From: Anak Subject: Re: Reading a Protractor

Donya Ina, I think that the easiest way to decide which scale to use is to look at the angle you're measuring. If it is greater than a right angle (more than ninety degrees) then use the scale that has numbers greater than 90. If the angle is obviously acute (measures less than 90 degrees), use the scale which has numbers less than 90. Of course you have to line up the protractor correctly with the center point on the vertex of the angle and one of the rays of the angle coincide with the line on the protractor.

I-push mo lang ‘yan!


Measure of an Angle

Wala Lang…

Two angles are congruent if they have the same measure.Symbol: “” (is congruent to)

Congruent Angles

B

A

C


Y

X

Z

ABC and XYZ are congruent, so we write

ABC XYZ which means mABC = mXYZ

Straight angle

m A= 180°

Obtuse angle

90 < m A < 180°

Acute angle

0< m A < 90°

Classifying Angles

Angles are classified as acute, right, obtuse, and straight, according to their measures. Angles have measures greater than 0°and less or equal to 180°.

A ARight angle

m A = 90°

A A


An acute angle is an angle measuring between 0 and 90 degrees.

Classifying Angles


An obtuse angle is an angle measuring between 90 and 180 degrees.

Classifying Angles


Classifying Angles


A right angle is an angle measuring exactly 90 degrees. It is formed by two perpendicular rays with a common endpoint. Denotes 90

degrees

Classifying Angles


A straight angle is an angle measuring exactly 180 degrees. It is formed by two opposite rays with a common

endpoint.

A

Determine whether each of the following is acute, right, or obtuse.

1.

2.

3.

4.

5.

??

acute obtuseright

obtuseacute


Complementary AnglesTwo angles are called complementary angles if the sum of their degree measurements equals 90 degrees.We say that one is the COMPLEMENT of the other.

Example: These two angles are complementary.

Angle Pairs


These two angles can be "pasted"

together to form a right angle!

Angle Pairs


Supplementary AnglesTwo angles are called supplementary angles if the sumof their degree measurements equals 180 degrees.

We say that one is the SUPPLEMENT of the other.

Example: These two angles are supplementary.

Angle Pairs


These two angles can be "pasted" together to

form a straight line!

Angle Pairs


1. Find the complement of 65.

Examples. (See solution on the board)

Answer : 25

Answer : 40

Angle Pairs


2. Two angles are supplementary. One measures 140. What is the measure of the other angle?

Answer : 30

3. The supplement of an angle is 30 more than twice its complement. What is the angle?

A. Determine the complement of

1. 48 2. 7942


11

B. Determine the supplement of

1. 48 2. 123132 57

Angle Pairs


Adjacent Angles

Adjacent angles are two angles with a common vertex and a common side but no common interior points.

A

B

C

DAdjacent angles:

BAC and DAB

Angle Pairs


Angle Addition PostulateThe sum of the measures of two adjacent angles is equal to the measure of the angle formed by the two nonadjacent rays.

A

B

C

D mBAC + mDAB

= mDAC

Angle Pairs


Linear PairsTwo adjacent angles form a linear pair if their uncommon sides form opposite rays

B

D

C

AABD and mDBC are linear pairs

Angle Pairs


Linear Pair PostulateLinear pairs are supplementary.

B

D

C

A

Since ABD and mDBC are linear pairs, then

they are supplementary; that

is, mABD + mDBC

= 180

Examples: Find x (see solution on the board).

2. 3.x

55 165x

x =35 x=15

Angle Pairs


1.

55x

P

U

S

H mPUH =140

x = 85

Examples:

Angle Pairs


3. The angles (3x – 20) and (7x) form a linear pair. What is the a. value of x? b. measure of each angle?

x = 20

40 and 140

Vertical Angles

VERTICAL angles are two nonadjacent angles formed by two intersecting lines.

Angle Pairs


AED and BEC are vertical angles

AEB and DEC are vertical angles

Vertical Angle TheoremVertical angles are congruent

Angle Pairs


AED and BEC are vertical angles, so AED BEC or mAED = mBEC

AEB and DEC are also vertical angles, so AEB DEC or mAEB = mDEC

Vertical AnglesNote that the adjacent angles form linear pairs.

Angle Pairs


The ff. are linear pairs:

AEB and BEC

AED and AEB

AED and CED

DEC and BEC

Example

1.

Angle Pairs


Example:

2. Find the value of x, y, and z.

58x

x = 58y = 122x = 122y

z

Angle Pairs


20

C

J

D

E F

G

H

70

9070

20

90

3. Given the figure shown on the right. Find the measures of the following:

)a DCE)b GCH)c ECF)d DCJ)e JCH

Angle Pairs


52

B

A

F E

D

C

60

G

68

68

6052

Check your understandingFind the measures of the numbered angles.

3 2

14

Summing it up!Angles & Angle Measurements

Thank you!

CABT Math 8 - Angles and Angle Measurements

Education

Transcript of CABT Math 8 - Angles and Angle Measurements