math 7 geometry 03 angles and angle measurements

The session shall begin shortly…

TXT, pp. 395-400, 406-416, 421-428, 432-435

An angle the union of two different rays that

have the same initial point (or common

endpoint). The rays are the sides of the angle.

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

Angles & Angle Measurements

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 in three different ways.

SRT, TRS, R

R

S

T

RS 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

Angles & Angle Measurements

Example: Here are some examples of angles

and their degree measurements.

Measure of an Angle

Angles & Angle Measurements

ProtractorA device used to measure angles.

Measure of an Angle

Angles & Angle Measurements

Measure of an Angle

Angles & Angle Measurements

A

B

C

What is the

measure of this

angle?

ANSWER: 50

Measure of an Angle

Angles & Angle Measurements

A

B

C

What is the

measure of this

angle?

ANSWER: 110

Measure of an Angle

Angles & Angle Measurements

A

B

C

What is the

measure of this

angle?

ANSWER: 90

Measure of an Angle

Angles & Angle Measurements

A

B

C

What is the

measure of this

angle?

ANSWER: 70

Reading a Protractor

Date: 01/13/2014 at 15:51:52

From: Donya Ina

Subject: 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!

Angles & Angle Measurements

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!

Angles & Angle Measurements

Measure of an Angle

Wala Lang…

The Angle

Measurement

postulate

To every angle, there corresponds a unique

measure r in degrees such that r is between

0 and 180.

Measure of an Angle

Angles & Angle Measurements

Measure of an Angle

B

A

C

In half-plane H1 through the endpoint of

ray BC lying in the edge of the half-

plane there is exactly one ray BA, with A

in H1, such that angle ABC formed by the

two rays has a given measure between 0

and 180 degrees.

Angles & Angle Measurements

The Angle construction Postulate (or Protractor

Postulate)

WHAT?!

What do the two postulates mean?

1] All angles are measured between 00 and 1800.

2] They can be measured with a protractor.

3] The measurement is the absolute values of the

numbers read on the protractor.

4] The values of 0 and 180 on the protractor

were arbitrarily selected.

Measure of an Angle

Angles & Angle Measurements

Two angles are congruent if they have the same

measure.

Symbol: “” (is congruent to)

Congruent Angles

B

A

C

Angles & Angle Measurements

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 A

Right angle

m A = 90°

A A

Angles & Angle Measurements

An acute angle is an angle measuring between 0

and 90 degrees.

Classifying Angles

Angles & Angle Measurements

An obtuse angle is an angle measuring between

90 and 180 degrees.

Classifying Angles

Angles & Angle Measurements

Classifying Angles

Angles & Angle Measurements

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

Angles & Angle Measurements

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

Check your understanding

Complementary Angles

Two 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

Angles & Angle Measurements

These two angles can

be "pasted" together to

form a right angle!

Angle Pairs

Angles & Angle Measurements

Supplementary AnglesTwo angles are called supplementary angles if the sum

of their degree measurements equals 180 degrees.

We say that one is the SUPPLEMENT of the other.

Example:

These two angles are supplementary.

Angle Pairs

Angles & Angle Measurements

These two angles can be

"pasted" together to form a

straight line!

Angle Pairs

Angles & Angle Measurements

1. Find the complement of 65.

Examples. (See solution on the board)

Answer : 25

Answer : 40

Angle Pairs

Angles & Angle Measurements

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 degrees more than

twice its complement. What is the angle?

A. Determine the complement of

1. 48 2. 7942

Check your understanding

11

B. Determine the supplement of

1. 48 2. 123132 57

Angle Pairs

Angles & Angle Measurements

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

Angles & Angle Measurements

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

Angles & Angle Measurements

Bisector of an AngleAn ANGLE BISECTOR is a ray in the interior of an angle

that divides the angle into two congruent adjacent angles

A

B

C

D

If ray AB is a bisector of

mDAC, then

BAC DAB

This means:

mBAC = mDAB

= ½ mDAC

Angle Pairs

Angles & Angle Measurements

Linear PairsTwo adjacent angles form a linear pair if their

uncommon sides form opposite rays

B

D

C

A

ABD and mDBC are

linear pairs

Angle Pairs

Angles & Angle Measurements

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 =35x=15

Angle Pairs

Angles & Angle Measurements

1.

55

x

P

U

S

H mPUH =140

x = 85

Examples: (see solution on the board).

Angle Pairs

Angles & Angle Measurements

4.

PS

H

a. If mHUP = 120, what is mSUH?

U

is a bisector of HUPUS

60

mHUS = 74

b. If mSUP = 74, what is mHUS and mPUH?

mPUH = 148

Vertical Angles

VERTICAL angles are two

nonadjacent angles

formed by two

intersecting lines.

Angle Pairs

Angles & Angle Measurements

AED and BEC are vertical angles

AEB and DEC are vertical angles

Vertical Angle TheoremVertical angles are

congruent

Angle Pairs

Angles & Angle Measurements

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

Angles & Angle Measurements

The ff. are linear pairs:

AEB and BEC

AED and AEB

AED and CED

DEC and BEC

Example

1.

Angle Pairs

Angles & Angle Measurements

Example:

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

58x

x = 58

y = 122

z = 122y

z

Angle Pairs

Angles & Angle Measurements

52

B

A

F E

D

C

60

G

68

68

60

52

Example 3. Find the measures of the numbered angles.

3 2

1

4

Angle Pairs

Angles & Angle Measurements

Perpendicular LinesLines that intersect at right

angles.

Angle Pairs

Angles & Angle Measurements

In the figure, line m is

perpendicular to line n.

We write: m n

m

n

Perpendicular BisectorA line, ray, or line segment

that passes through the

midpoint of a segment.

Angle Pairs

Angles & Angle Measurements

In the figure, line m is a

perpendicular bisector of segment

AB. It means:

• m is perpendicular to line AB

• m passes through the midpoint C,

which means AC = CB

m

A BC

D

20

C

J

D

EF

G

H

70

9070

20

90

Example:

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

Angles & Angle Measurements

Summing it up!Angles & Angle Measurements

Thank you!