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

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

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

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

5. 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

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

7. Straight Angles Opposite rays
___________ are two rays that are part of a the same line and have only their endpoints in common. X Y Z
opposite rays
XY and XZ are ____________.
The figure formed by opposite rays is also referred to as a ____________. A straight angle measures 180 degrees.
straight angle

8. Congruent Angles measure
Recall that congruent segments have the same ________.
measure
Congruent angles
_______________ also have the same measure.

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

10. Congruent Angles To show that 1 is congruent to 2, we use ____. arcs
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

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

12. Angle Pair Relationships

13. Angle Pair Relationship Essential Questions
How are special angle pairs identified?

14. Adjacent Angles When you “split” an angle, you create two angles.
The two angles are called
_____________ A C B
adjacent angles D
adjacent = next to, joining. 2 1
1 and 2 are examples of adjacent angles. They share a common ray.
Name the ray that 1 and 2 have in common. ____

15. Adjacent Angles Adjacent angles are angles that:
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

16. Adjacent 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 ____

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

18. Vertical Angles Two angles are vertical if they are two
Definition of
Vertical
Angles
Two angles are vertical if they are two
nonadjacent angles formed by a pair of
intersecting lines.
Vertical angles:
1 and 3 1 4 2
2 and 4 3

19. Vertical Angles Vertical angles are congruent. 1  3 2  4
Theorem 3-1
Vertical Angle
Theorem
Vertical angles are congruent. n m 2
1  3 3 1
2  4 4

20. Vertical Angles Find the value of x in the figure:
130° x°
The angles are vertical angles.
So, the value of x is 130°.

21. 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
linear pair
These angles are called a _________

22. Linear Pairs of Angles Note: Two angles form a linear pair if:
Definition of
Linear Pairs
Two angles form a linear pair if:
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:

23. 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.

24. Complementary Angles Two angles are complementary if
Definition of
Complementary
Angles
Two angles are complementary if
the sum of their degree measure is 90.
60° D E F
30° A B C
mABC + mDEF = = 90

25. Complementary 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.

26. 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

27. 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 the sum of their degree measure is 180.
130° D E F
50° A B C
mABC + mDEF = = 180

28. 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

29. Vertical 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.

30. 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°

31. Congruent Angles A B C D E G H 1 2 3 4
1) If m1 = 2x + 3 and the m3 = 3x + 2, 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°