1. Angles § 3.1 Angles § 3.2 Angle Measure
§ 3.3 The Angle Addition Postulate
§ 3.4 Adjacent Angles and Linear Pairs of Angles
§ 3.5 Complementary and Supplementary Angles
§ 3.6 Congruent Angles
§ 3.7 Perpendicular Lines

2. Vocabulary What You'll Learn
Angles
What You'll Learn
You will learn to name and identify parts of an angle.
Vocabulary
1) Opposite Rays
2) Straight Angle
3) Angle
4) Vertex
5) Sides
6) Interior
7) Exterior

3. XY and XZ are ____________. opposite rays
Angles
Opposite rays
___________ are two rays that are part of a the same line and have only their endpoints in common. X Y Z
XY and XZ are ____________.
opposite rays
The figure formed by opposite rays is also referred to as a ____________.
straight angle

4. There is another case where two rays can have a common endpoint.
Angles
There is another case where two rays can have a common endpoint.
This figure is called an _____.
angle
Some parts of angles have special names. S
The common endpoint is called the ______,
vertex
and the two rays that make up the sides of the angle are called the sides of the angle.
side R T
side
vertex

5. There are several ways to name this angle.
Angles
There are several ways to name this angle.
1) Use the vertex and a point from each side.
SRT or
TRS S
The vertex letter is always in the middle.
2) Use the vertex only.
side R
If there is only one angle at a vertex, then the angle can be named with that vertex. 1 R T
side
vertex
3) Use a number. 1

6. Angles
Definition of Angle
An angle is a figure formed by two noncollinear rays that have a common endpoint. E D F 2
Symbols:
DEF
FED E 2

7. 1) Name the angle in four ways.
Angles
1) Name the angle in four ways. B A 1 C
ABC
CBA B 1
2) Identify the vertex and sides of this angle.
vertex:
Point B
sides:
BA and BC

8. 1) Name all angles having W as their vertex.2 1 W 2
XWZ Y
2) What are other names for ? 1 Z
XWY or
YWX
3) Is there an angle that can be named ? W
No!

9. An angle separates a plane into three parts:
1) the ______
interior
exterior
2) the ______
exterior W Y Z A
interior
3) the _________
angle itself
In the figure shown, point B and all other points in the blue region are in the interior of the angle. B
Point A and all other points in the green region are in the exterior of the angle.
Points Y, W, and Z are on the angle.

10. Angles
Is point B in the interior of the angle, exterior of the angle, or on the angle? P G
Exterior B
Is point G in the interior of the angle, exterior of the angle, or on the angle?
On the angle
Is point P in the interior of the angle, exterior of the angle, or on the angle?
Interior

11. Vocabulary What You'll Learn
§3.2 Angle Measure
What You'll Learn
You will learn to measure, draw, and classify angles.
Vocabulary
1) Degrees
2) Protractor
3) Right Angle
4) Acute Angle
5) Obtuse Angle

12. In geometry, angles are measured in units called _______. degrees
§3.2 Angle Measure
In geometry, angles are measured in units called _______.
degrees
The symbol for degree is °. Q P R
75°
In the figure to the right, the angle is 75 degrees.
In notation, there is no degree symbol with 75
because the measure of an angle is a real
number with no unit of measure.
m PQR = 75

13. Angles Measure Postulate
§3.2 Angle Measure
Postulate 3-1
Angles Measure Postulate
For every angle, there is a unique positive number between __ and ____ called the degree measure of the angle.
180 B A C n°
m ABC = n
and 0 < n < 180

14. Use a protractor to measure SRQ.
§3.2 Angle Measure
You can use a _________ to measure angles and sketch angles of given measure.
protractor
Use a protractor to measure SRQ.
1) Place the center point of the protractor on vertex R Align the straightedge with side RS. Q R S
2) Use the scale that begins with at RS Read where the other side of the angle, RQ, crosses this scale.

15. Find the measurement of: m SRH 70
§3.2 Angle Measure
Find the measurement of:
m SRH 70
m SRQ =
180
m QRG =
180 – 150
= 30
m SRJ = 45
m GRJ =
150 – 45
= 105
m SRG =
150 H J G S Q R

16. Use a protractor to draw an angle having a measure of 135.
§3.2 Angle Measure
Use a protractor to draw an angle having a measure of 135.
1) Draw AB
3) Locate and draw point C at the mark labeled Draw AC.
2) Place the center point of the protractor on A. Align the mark labeled 0 with the ray. C A B

17. obtuse angle 90 < m A < 180 right angle m A = 90
§3.2 Angle Measure
Once the measure of an angle is known, the angle can be classified as one of three types of angles. These types are defined in relation to a right angle.
Types of Angles A A A
obtuse angle 90 < m A < 180
right angle m A = 90
acute angle 0 < m A < 90

18. Classify each angle as acute, obtuse, or right.
§3.2 Angle Measure
Classify each angle as acute, obtuse, or right.
110°
90°
40°
Obtuse
Right
Acute
75°
50°
130°
Acute
Obtuse
Acute

19. The measure of B is 138. Solve for x.
§3.2 Angle Measure
5x - 7 B
The measure of B is 138. Solve for x.
9y + 4 H
The measure of H is 67. Solve for y.
Given: (What do you know?)
Given: (What do you know?)
B = 5x – 7 and B = 138
H = 9y and H = 67
5x – 7 = 138
9y + 4 = 67
Check!
Check!
5x = 145
9y = 63
5(29) -7 = ?
9(7) + 4 = ?
x = 29
y = 7
= ?
= ?
138 = 138
67 = 67

20. ? ? ?
Is m a larger than m b ?
60°
60°

21. End of Lesson

22. §3.3 The Angle Addition Postulate
What You'll Learn
You will learn to find the measure of an angle and the bisector of an angle.
Vocabulary
NOTHING NEW!

23. §3.3 The Angle Addition Postulate
1) Draw an acute, an obtuse, or a right angle Label the angle RST. R T S
45°
2) Draw and label a point X in the interior of the angle. Then draw SX. X
75°
30°
3) For each angle, find mRSX, mXST, and RST.

24. §3.3 The Angle Addition Postulate
1) How does the sum of mRSX and mXST compare to mRST ?
Their sum is equal to the measure of RST .
mXST = 30
+ mRSX = 45
= mRST = 75 R T S
2) Make a conjecture about the relationship between the two smaller angles and the larger angle.
45° X
The sum of the measures of the two smaller angles is equal to the measure of the larger angle.
The Angle Addition Postulate (Video)
75°
30°

25. §3.3 The Angle Addition Postulate Angle Addition Postulate
For any angle PQR, if A is in the interior of PQR, then mPQA + mAQR = mPQR. 2 1 A R P Q
m1 + m2 = mPQR.
There are two equations that can be derived using Postulate 3 – 3.
m1 = mPQR – m2
These equations are true no matter where A is located in the interior of PQR.
m2 = mPQR – m1

26. §3.3 The Angle Addition Postulate
Find m2 if mXYZ = 86 and m1 = 22. 2 1 Y Z X W
m2 + m1 = mXYZ
Postulate 3 – 3.
m2 = mXYZ – m1
m2 = 86 – 22
m2 = 64

27. §3.3 The Angle Addition Postulate
Find mABC and mCBD if mABD = 120.
mABC + mCBD = mABD
Postulate 3 – 3.
2x + (5x – 6) = 120
Substitution
7x – 6 = 120
Combine like terms
7x = 126
Add 6 to both sides
x = 18
Divide each side by 7
= 120
2x°
(5x – 6)° B D C A
mABC = 2x
mCBD = 5x – 6
mABC = 2(18)
mCBD = 5(18) – 6
mABC = 36
mCBD = 90 – 6
mCBD = 84

28. §3.3 The Angle Addition Postulate
Just as every segment has a midpoint that bisects the segment, every angle
has a ___ that bisects the angle.
ray
This ray is called an ____________ .
angle bisector

29. §3.3 The Angle Addition Postulate
Definition of
an Angle Bisector
The bisector of an angle is the ray with its endpoint at the vertex of the angle, extending into the interior of the angle.
The bisector separates the angle into two angles of equal measure. 2 1 A R P Q
is the bisector of PQR.
m1 = m2

30. §3.3 The Angle Addition Postulate
If bisects CAN and mCAN = 130, find 1 and 2.
Since bisects CAN, 1 = 2. 1 2 A C N T
1 + 2 = CAN
Postulate 3 - 3
1 + 2 = 130
Replace CAN with 130
1 + 1 = 130
Replace 2 with 1
2(1) = 130
Combine like terms
(1) = 65
Divide each side by 2
Since 1 = 2, 2 = 65

31. End of Lesson

32. Adjacent Angles and Linear Pairs of Angles
What You'll Learn
You will learn to identify and use adjacent angles and linear pairs of angles. A C B
When you “split” an angle, you create two angles. D
The two angles are called
_____________
adjacent angles 2 1
adjacent = next to, joining.
1 and 2 are examples of adjacent angles. They share a common ray.
Name the ray that 1 and 2 have in common. ____

33. Adjacent Angles and Linear Pairs of Angles
Definition of
Adjacent
Angles
Adjacent angles are angles that:
A) share a common side
B) have the same vertex, and
C) have no interior points in common M J N R 1 2
1 and 2 are adjacent
with the same vertex R and
common side

34. Adjacent Angles and Linear Pairs of Angles
Determine whether 1 and 2 are adjacent angles.
No. They have a common vertex B, but
_____________ 1 2 B
no common side
Yes. They have the same vertex G and a common side with no interior points in common. 1 2 G N 1 2 J L
No. They do not have a common vertex or ____________
a common side
The side of 1 is ____
The side of 2 is ____

35. Adjacent Angles and Linear Pairs of Angles
Determine whether 1 and 2 are adjacent angles.
No. 1 2
Yes. 1 2 X D Z
In this example, the noncommon sides of the adjacent angles form a
___________.
straight line
These angles are called a _________
linear pair

36. Adjacent Angles and Linear Pairs of Angles
Definition of
Linear Pairs
Two angles form a linear pair if and only if (iff):
A) they are adjacent and
B) their noncommon sides are opposite rays C A D B 1 2
1 and 2 are a linear pair.
Note:

37. Adjacent Angles and Linear Pairs of Angles
In the figure, and are opposite rays. 1 2 M 4 3 E H T A C
1) Name the angle that forms a
linear pair with 1.
ACE
ACE and 1 have a common side ,
the same vertex C, and opposite rays
and
2) Do 3 and TCM form a linear pair? Justify your answer.
No. Their noncommon sides are not opposite rays.

38. End of Lesson

39. §3.5 Complementary and Supplementary Angles
What You'll Learn
You will learn to identify and use Complementary and
Supplementary angles

40. §3.5 Complementary and Supplementary Angles
Definition of
Complementary
Angles
Two angles are complementary if and only if (iff) the sum of their degree measure is 90.
60° D E F
30° A B C
mABC + mDEF = = 90

41. §3.5 Complementary and Supplementary Angles
If two angles are complementary, each angle is a complement of the other.
ABC is the complement of DEF and DEF is the complement of ABC.
60° D E F
30° A B C
Complementary angles DO NOT need to have a common side or even the
same vertex.

42. §3.5 Complementary and Supplementary Angles
Some examples of complementary angles are shown below.
75° I
mH + mI = 90
15° H
50° H
40° Q P S
mPHQ + mQHS = 90
30°
60° T U V W Z
mTZU + mVZW = 90

43. §3.5 Complementary and Supplementary Angles
If the sum of the measure of two angles is 180, they form a special pair of
angles called supplementary angles.
Definition of
Supplementary
Angles
Two angles are supplementary if and only if (iff) the sum of their degree measure is 180.
130° D E F
50° A B C
mABC + mDEF = = 180

44. §3.5 Complementary and Supplementary Angles
Some examples of supplementary angles are shown below.
105° H
75° I
mH + mI = 180
50° H
130° Q P S
mPHQ + mQHS = 180
60°
120° T U V W Z
mTZU + mUZV = 180
and
mTZU + mVZW = 180

45. End of Lesson

46. What You'll Learn You will learn to identify and use congruent and
§3.6 Congruent Angles
What You'll Learn
You will learn to identify and use congruent and
vertical angles.
Recall that congruent segments have the same ________.
measure
_______________ also have the same measure.
Congruent angles

47. Two angles are congruent iff, they have the same ______________.
§3.6 Congruent Angles
Definition of
Congruent
Angles
Two angles are congruent iff, they have the same
______________.
degree measure
B  V iff
50° V
mB = mV
50° B

48. To show that 1 is congruent to 2, we use ____. arcs
§3.6 Congruent Angles
To show that 1 is congruent to 2, we use ____.
arcs 1 2
To show that there is a second set of congruent angles, X and Z, we use double arcs.
This “arc” notation states that: Z X
X  Z
mX = mZ

49. When two lines intersect, ____ angles are formed. four
§3.6 Congruent Angles
When two lines intersect, ____ angles are formed.
four
There are two pair of nonadjacent angles.
These pairs are called _____________.
vertical angles 1 4 2 3

50. Two angles are vertical iff they are two nonadjacent
§3.6 Congruent Angles
Definition of
Vertical
Angles
Two angles are vertical iff they are two nonadjacent
angles formed by a pair of intersecting lines.
Vertical angles:
1 and 3 1 4 2
2 and 4 3

51. Hands-On 1) On a sheet of paper, construct two intersecting lines
§3.6 Congruent Angles
1) On a sheet of paper, construct two intersecting lines
that are not perpendicular.
2) With a protractor, measure each angle formed.
3) Make a conjecture about vertical angles. 1 2 3 4
Consider:
A. 1 is supplementary to 4.
m1 + m4 = 180
Hands-On
B. 3 is supplementary to 4.
m3 + m4 = 180
Therefore, it can be shown that
1  3
Likewise, it can be shown that
2  4

52. 1) If m1 = 4x + 3 and the m3 = 2x + 11, then find the m3
§3.6 Congruent Angles 1 2 3 4
1) If m1 = 4x + 3 and the m3 = 2x + 11, then find the m3
x = 4; 3 = 19°
2) If m2 = x + 9 and the m3 = 2x + 3, then find the m4
x = 56; 4 = 65°
3) If m2 = 6x - 1 and the m4 = 4x + 17, then find the m3
x = 9; 3 = 127°
4) If m1 = 9x - 7 and the m3 = 6x + 23, then find the m4
x = 10; 4 = 97°

53. Vertical angles are congruent.
§3.6 Congruent Angles
Theorem 3-1
Vertical Angle
Theorem
Vertical angles are congruent. n m 2
1  3 3 1
2  4 4

54. Find the value of x in the figure:
§3.6 Congruent Angles
Find the value of x in the figure:
The angles are vertical angles.
130° x°
So, the value of x is 130°.

55. Find the value of x in the figure:
§3.6 Congruent Angles
Find the value of x in the figure:
The angles are vertical angles.
(x – 10) = 125.
(x – 10)°
x – 10 = 125.
125°
x = 135.

56. Suppose two angles are congruent.
§3.6 Congruent Angles
Suppose two angles are congruent.
What do you think is true about their complements?
1  2
1 + x = 90
2 + y = 90
x is the complement
of 1
y is the complement
of 2
x = 90 - 1
y = 90 - 2
Because 1  2, a “substitution” is made.
x = 90 - 1
y = 90 - 1
x = y
x  y
If two angles are congruent, their complements are congruent.

57. If two angles are congruent, then their complements are _________.
§3.6 Congruent Angles
Theorem 3-2
If two angles are congruent, then their complements are
_________.
congruent
The measure of angles complementary to A and B
is 30. A B
60°
A  B
Theorem 3-3
If two angles are congruent, then their supplements are
_________.
congruent
The measure of angles supplementary to 1 and 4
is 110. 4 3 2 1
70°
110°
110°
70°
4  1

58. If two angles are complementary to the same angle,
§3.6 Congruent Angles
Theorem 3-4
If two angles are complementary to the same angle,
then they are _________.
congruent
3 is complementary to 4
5 is complementary to 4 4 3
5  3 5
Theorem 3-5
If two angles are supplementary to the same angle,
then they are _________.
congruent 3 1 2
1 is supplementary to 2
3 is supplementary to 2
1  3

59. Find the measure of an angle that is supplementary to B.
§3.6 Congruent Angles
Suppose A  B and mA = 52.
Find the measure of an angle that is supplementary to B. A
52° B
52° 1
B + 1 = 180
1 = 180 – B
1 = 180 – 52
1 = 128°

60. If 1 is complementary to 3, 2 is complementary to 3, and m3 = 25,
§3.6 Congruent Angles
If 1 is complementary to 3,
2 is complementary to 3,
and m3 = 25,
What are m1 and m2 ?
m1 + m3 = Definition of complementary angles.
m1 = 90 - m Subtract m3 from both sides.
m1 = Substitute 25 in for m3.
m1 = Simplify the right side.
You solve for m2
m2 + m3 = Definition of complementary angles.
m2 = 90 - m Subtract m3 from both sides.
m2 = Substitute 25 in for m3.
m2 = Simplify the right side.

61. 1) If m1 = 2x + 3 and the m3 = 3x - 14, then find the m3
§3.6 Congruent Angles A B C D E G H 1 2 3 4
1) If m1 = 2x + 3 and the m3 = 3x - 14, then find the m3
x = 17; 3 = 37°
2) If mABD = 4x + 5 and the mDBC = 2x + 1, then find the mEBC
x = 29; EBC = 121°
3) If m1 = 4x and the m3 = 2x + 19, then find the m4
x = 16; 4 = 39°
4) If mEBG = 7x and the mEBH = 2x + 7, then find the m1
x = 18; 1 = 43°

62. Suppose you draw two angles that are congruent and supplementary.
What is true about the angles?

63. All right angles are _________. congruent
§3.6 Congruent Angles
Theorem 3-6
If two angles are congruent and supplementary then each is a __________.
right angle
1 is supplementary to 2 1 2
1 and 2 = 90
Theorem 3-7
All right angles are _________.
congruent C B A
A  B  C

64. If 1 is supplementary to 4, 3 is supplementary to 4, and
§3.6 Congruent Angles
If 1 is supplementary to 4, 3 is supplementary to 4, and
m 1 = 64, what are m 3 and m 4? A D C B E 1 2 3 4
1  3
They are vertical angles.
m 1 = m3
m 3 = 64
3 is supplementary to 4
Given
m3 + m4 = 180
Definition of supplementary.
64 + m4 = 180
m4 = 180 – 64
m4 = 116

65. End of Lesson

66. §3.7 Perpendicular Lines
What You'll Learn
You will learn to identify, use properties of, and construct
perpendicular lines and segments.

67. Lines that intersect at an angle of 90 degrees are _________________.
§3.7 Perpendicular Lines
Lines that intersect at an angle of 90 degrees are _________________.
perpendicular lines
In the figure below, lines are perpendicular. A D C B 1 2 3 4

68. Perpendicular lines are lines that intersect to form a right angle.
Definition of
Perpendicular
Lines
Perpendicular lines are lines that intersect to form a
right angle. m n

69. In the figure below, l  m. The following statements are true.
§3.7 Perpendicular Lines
In the figure below, l  m. The following statements are true. m 2 1 3 4 l
1) 1 is a right angle.
Definition of Perpendicular Lines
2) 1  3.
Vertical angles are congruent
3) 1 and 4 form a linear pair.
Definition of Linear Pair
4) 1 and 4 are supplementary.
Linear pairs are supplementary
5) 4 is a right angle.
m = 180, m4 = 90
6) 2 is a right angle.
Vertical angles are congruent

70. If two lines are perpendicular, then they form four right angles.
§3.7 Perpendicular Lines
Theorem 3-8
If two lines are perpendicular, then they form four right angles. 1 3 4 2 a b

71. 1) PRN is an acute angle. False. 2) 4  8 True
§3.7 Perpendicular Lines
1) PRN is an acute angle.
False.
2) 4  8
True

72. If a line m is in a plane and T is a point in m, then there
§3.7 Perpendicular Lines
Theorem 3-9
If a line m is in a plane and T is a point in m, then there
exists exactly ___ line in that plane that is perpendicular to
m at T.
one m T

73. End of Lesson