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Angles & Angle Measurements

Angles
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.
vertex

B
sides

A
C
Angles & Angle Measurements

Naming Angles
B

A

1

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

C

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
Check your understanding
Use the diagram below to answer the following questions.
a. Name the vertex.
R

S

b. Name the sides of the angle.

RT and RS
c. Name the angle three different ways.
SRT,  TRS,  R

R
T
Angles & Angle Measurements

Measure of an Angle
B

A
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:
mBAC, mCAB, mA
Angles & Angle Measurements

Measure of an Angle
Example: Here are some examples of angles
and their degree measurements.
Angles & Angle Measurements

Measure of an Angle

Protractor
A device used to measure angles.
Angles & Angle Measurements

Measure of an Angle
What is the
measure of this
angle?

B

A

C

ANSWER: 50
Angles & Angle Measurements

Measure of an Angle
What is the
measure of this
angle?

B

A

C

ANSWER: 110
Angles & Angle Measurements

Measure of an Angle
What is the
measure of this
angle?

B

A

C

ANSWER: 90
Angles & Angle Measurements

Measure of an Angle
What is the
measure of this
angle?

B

C

A

ANSWER: 70
Angles & Angle Measurements

Measure of an Angle
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 magmeasure 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!
Wala Lang…
Angles & Angle Measurements

Congruent Angles
Two angles are congruent if they have the same
measure.
Symbol: “” (is congruent to)
A
B

X
Y

C

Z
ABC and XYZ are congruent, so we write
ABC  XYZ which means mABC = mXYZ
Angles & Angle Measurements

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

A

Acute angle

Right angle

Obtuse angle

Straight angle

0< m A < 90°

m A = 90°

90 < m A < 180°

m A= 180°

A
Angles & Angle Measurements

Classifying Angles
An acute angle is an angle measuring between 0
and 90 degrees.
Angles & Angle Measurements

Classifying Angles
An obtuse angle is an angle measuring between
90 and 180 degrees.
Angles & Angle Measurements

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
Angles & Angle Measurements

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

A
Check your understanding
Determine whether each of the following is
acute, right, or obtuse.
1.
acute

2.

3.

5.
obtuse

right

4.
?

acute

?

obtuse
Angles & Angle Measurements

Angle Pairs
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.
Angles & Angle Measurements

Angle Pairs
These two angles can
be "pasted" together to
form a right angle!
Angles & Angle Measurements

Angle Pairs
Supplementary Angles
Two 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.
Angles & Angle Measurements

Angle Pairs

These two angles can be
"pasted" together to form a
straight line!
Angles & Angle Measurements

Angle Pairs
Examples. (See solution on the board)
1. Find the complement of 65.
Answer : 25
2. Two angles are supplementary. One measures 140.
What is the measure of the other angle?
Answer : 40
3. The supplement of an angle is 30 more than twice
its complement. What is the angle?

Answer : 30
Check your understanding
A. Determine the complement of
1. 48
42
2. 79
11
B. Determine the supplement of
1. 48
132
2. 123 57
Angles & Angle Measurements

Angle Pairs
Adjacent Angles
Adjacent angles are two angles with a common vertex
and a common side but no common interior points.
D

B

Adjacent angles:

BAC and DAB

A
C
Angles & Angle Measurements

Angle Pairs
Angle Addition Postulate
The sum of the measures of two adjacent angles is
equal to the measure of the angle formed by the two
nonadjacent rays.
D

B

mBAC + mDAB

= mDAC
A
C
Angles & Angle Measurements

Angle Pairs
Linear Pairs
Two adjacent angles form a linear pair if their
uncommon sides form opposite rays
D
A

B
C

ABD and mDBC are
linear pairs
Angles & Angle Measurements

Angle Pairs
Linear Pair Postulate
Linear pairs are supplementary.
D
A

B
C

Since ABD and mDBC
are linear pairs, then
they are supplementary;
that is,

mABD + mDBC =
180
Angles & Angle Measurements

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

U

1.

mPUH =140

x

P
2.

55

x = 85

S
3.

x
55
x =35

x

165
x=15
Angles & Angle Measurements

Angle Pairs
Examples:
3. The angles (3x – 20) and (7x) form a linear pair.
What is the
x = 20
a. value of x?
b. measure of each angle? 40 and 140
Angles & Angle Measurements

Angle Pairs
Vertical Angles
VERTICAL angles are two
nonadjacent angles
formed by two
intersecting lines.
AED and BEC are vertical angles
AEB and DEC are vertical angles
Angles & Angle Measurements

Angle Pairs
Vertical Angle Theorem
Vertical angles are
congruent
AED and BEC are vertical
angles, so AED  BEC or
mAED = mBEC
AEB and DEC are also vertical
angles, so AEB  DEC or
mAEB = mDEC
Angles & Angle Measurements

Angle Pairs
Vertical Angles
Note that the adjacent
angles form linear pairs.
The ff. are linear pairs:
AEB and BEC
AED and AEB
AED and CED
DEC and BEC
Angles & Angle Measurements

Angle Pairs
Example
1.
Angles & Angle Measurements

Angle Pairs
Example:
2. Find the value of x, y, and z.
z
x

58
y

x = 58
y = 122
x = 122
Angles & Angle Measurements

Angle Pairs
3. Given the figure shown
on the right. Find the
measures of the following:
a)
b)
c)
d)
e)

DCE
GCH
ECF
DCJ
JCH

F

E
20

70

90

D

C

70
J

G
90

20
H
Check your understanding
Find the measures of the numbered angles.
68
B
C
1
52

A

3
60

G
68

4

60
D

2
52

F

E
Angles & Angle Measurements

Summing it up!
Thank you!

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Angle Measurements Session Begins

  • 1. The session shall begin shortly…
  • 2.
  • 3.
  • 4. Angles & Angle Measurements Angles 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. vertex B sides A C
  • 5. Angles & Angle Measurements Naming Angles B A 1 Notation: We denote an angle with three points and “” symbol. Naming: C 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
  • 6. Check your understanding Use the diagram below to answer the following questions. a. Name the vertex. R S b. Name the sides of the angle. RT and RS c. Name the angle three different ways. SRT,  TRS,  R R T
  • 7. Angles & Angle Measurements Measure of an Angle B A 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: mBAC, mCAB, mA
  • 8. Angles & Angle Measurements Measure of an Angle Example: Here are some examples of angles and their degree measurements.
  • 9. Angles & Angle Measurements Measure of an Angle Protractor A device used to measure angles.
  • 10. Angles & Angle Measurements Measure of an Angle What is the measure of this angle? B A C ANSWER: 50
  • 11. Angles & Angle Measurements Measure of an Angle What is the measure of this angle? B A C ANSWER: 110
  • 12. Angles & Angle Measurements Measure of an Angle What is the measure of this angle? B A C ANSWER: 90
  • 13. Angles & Angle Measurements Measure of an Angle What is the measure of this angle? B C A ANSWER: 70
  • 14. Angles & Angle Measurements Measure of an Angle 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 magmeasure 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!
  • 15. 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!
  • 17. Angles & Angle Measurements Congruent Angles Two angles are congruent if they have the same measure. Symbol: “” (is congruent to) A B X Y C Z ABC and XYZ are congruent, so we write ABC  XYZ which means mABC = mXYZ
  • 18. Angles & Angle Measurements 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 A Acute angle Right angle Obtuse angle Straight angle 0< m A < 90° m A = 90° 90 < m A < 180° m A= 180° A
  • 19. Angles & Angle Measurements Classifying Angles An acute angle is an angle measuring between 0 and 90 degrees.
  • 20. Angles & Angle Measurements Classifying Angles An obtuse angle is an angle measuring between 90 and 180 degrees.
  • 21. Angles & Angle Measurements 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
  • 22. Angles & Angle Measurements Classifying Angles A straight angle is an angle measuring exactly 180 degrees. It is formed by two opposite rays with a common endpoint. A
  • 23. Check your understanding Determine whether each of the following is acute, right, or obtuse. 1. acute 2. 3. 5. obtuse right 4. ? acute ? obtuse
  • 24. Angles & Angle Measurements Angle Pairs 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.
  • 25. Angles & Angle Measurements Angle Pairs These two angles can be "pasted" together to form a right angle!
  • 26. Angles & Angle Measurements Angle Pairs Supplementary Angles Two 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.
  • 27. Angles & Angle Measurements Angle Pairs These two angles can be "pasted" together to form a straight line!
  • 28. Angles & Angle Measurements Angle Pairs Examples. (See solution on the board) 1. Find the complement of 65. Answer : 25 2. Two angles are supplementary. One measures 140. What is the measure of the other angle? Answer : 40 3. The supplement of an angle is 30 more than twice its complement. What is the angle? Answer : 30
  • 29. Check your understanding A. Determine the complement of 1. 48 42 2. 79 11 B. Determine the supplement of 1. 48 132 2. 123 57
  • 30. Angles & Angle Measurements Angle Pairs Adjacent Angles Adjacent angles are two angles with a common vertex and a common side but no common interior points. D B Adjacent angles: BAC and DAB A C
  • 31. Angles & Angle Measurements Angle Pairs Angle Addition Postulate The sum of the measures of two adjacent angles is equal to the measure of the angle formed by the two nonadjacent rays. D B mBAC + mDAB = mDAC A C
  • 32. Angles & Angle Measurements Angle Pairs Linear Pairs Two adjacent angles form a linear pair if their uncommon sides form opposite rays D A B C ABD and mDBC are linear pairs
  • 33. Angles & Angle Measurements Angle Pairs Linear Pair Postulate Linear pairs are supplementary. D A B C Since ABD and mDBC are linear pairs, then they are supplementary; that is, mABD + mDBC = 180
  • 34. Angles & Angle Measurements Angle Pairs Examples: Find x (see solution on the board). H U 1. mPUH =140 x P 2. 55 x = 85 S 3. x 55 x =35 x 165 x=15
  • 35. Angles & Angle Measurements Angle Pairs Examples: 3. The angles (3x – 20) and (7x) form a linear pair. What is the x = 20 a. value of x? b. measure of each angle? 40 and 140
  • 36. Angles & Angle Measurements Angle Pairs Vertical Angles VERTICAL angles are two nonadjacent angles formed by two intersecting lines. AED and BEC are vertical angles AEB and DEC are vertical angles
  • 37. Angles & Angle Measurements Angle Pairs Vertical Angle Theorem Vertical angles are congruent AED and BEC are vertical angles, so AED  BEC or mAED = mBEC AEB and DEC are also vertical angles, so AEB  DEC or mAEB = mDEC
  • 38. Angles & Angle Measurements Angle Pairs Vertical Angles Note that the adjacent angles form linear pairs. The ff. are linear pairs: AEB and BEC AED and AEB AED and CED DEC and BEC
  • 39. Angles & Angle Measurements Angle Pairs Example 1.
  • 40. Angles & Angle Measurements Angle Pairs Example: 2. Find the value of x, y, and z. z x 58 y x = 58 y = 122 x = 122
  • 41. Angles & Angle Measurements Angle Pairs 3. Given the figure shown on the right. Find the measures of the following: a) b) c) d) e) DCE GCH ECF DCJ JCH F E 20 70 90 D C 70 J G 90 20 H
  • 42. Check your understanding Find the measures of the numbered angles. 68 B C 1 52 A 3 60 G 68 4 60 D 2 52 F E
  • 43. Angles & Angle Measurements Summing it up!