1. Warm up 9/15/14 Answer the question and draw a picture if you can:
1) What is a right angle?
2) What is an acute angle?
3) What is an obtuse angle?
4) What is a vertical angle?

2. Angle Pair Relationships

3. Angle Pair Relationship Essential Questions
How do I prove geometric theorems involving lines, angles?
Standard: MCC9-12.G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

4. Straight Angles Opposite rays
___________ are two rays that are part of 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

5. 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

6. 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

7. 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

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

9. 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. ____

10. 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

11. 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 ____

12. Linear Pairs of Angles Note:
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:

13. 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.

14. 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

15. Vertical Angles Two angles are vertical iff they are two
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

16. 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

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

18. 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.

19. 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

20. 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.

21. 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

22. 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

23. 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

24. 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°

25. 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°