Warm up 9/2/14
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Warm up 9/2/14 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?
Angle Pair Relationships
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.
Straight AnglesStraight Angles
___________ are two rays that are part of a the same line and have only their endpoints in common.
Opposite rays
XY Z
XY and XZ are ____________.opposite rays
The figure formed by opposite rays is also referred to as a ____________. A straight angle measures 180 degrees.straight angle
Naming AnglesNaming Angles
R
S
T
vertex
side
side
There are several ways to name this angle.
1) Use the vertex and a point from each side.
SRT or TRS
The vertex letter is always in the middle.
2) Use the vertex only.
R
If there is only one angle at a vertex, then theangle can be named with that vertex.
3) Use a number.
1
1
AnglesAngles
Definitionof Angle
An angle is a figure formed by two noncollinear rays that have a common endpoint.
E
D
F
2
Symbols: DEF
2
E
FED
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
right angle m A = 90
acute angle 0 < m A < 90
A
obtuse angle 90 < m A < 180
A
Angle MeasureAngle Measure
Classify each angle as acute, obtuse, or right.
110°
90°40°
50°
130° 75°
Obtuse
Obtuse
Acute
Acute Acute
Right
Angle MeasureAngle Measure
Adjacent Angles Adjacent Angles
When you “split” an angle, you create two angles.
D
A
C
B1
2
The two angles are called _____________adjacent angles
1 and 2 are examples of adjacent angles. They share a common ray.
Name the ray that 1 and 2 have in common. ____BD
adjacent = next to, joining.
Adjacent Angles Adjacent Angles
Definition of
Adjacent
Angles
Adjacent angles are angles that:
M
J
N
R1
21 and 2 are adjacent
with the same vertex R and
common side RM
A) share a common side
B) have the same vertex, and
C) have no interior points in common
Adjacent AnglesAdjacent Angles
Determine whether 1 and 2 are adjacent angles.
No. They have a common vertex B, but _____________no common side
1 2
B
12
G
Yes. They have the same vertex G and a common side with no interior points in common.
N
1
2J
L
No. They do not have a common vertex or ____________a common side
The side of 1 is ____LN
JNThe side of 2 is ____
Linear Pairs of AnglesLinear Pairs of Angles
Definition of
Linear Pairs
Two angles form a linear pair if and only if (iff):
1 and 2 are a linear pair.
A) they are adjacent and
B) their noncommon sides are opposite rays
C
A DB
1 2
AD form and BDBA
180 2 1
Linear Pairs of AnglesLinear Pairs of AnglesIn the figure, and are opposite rays.CM CE
1
2
M
43 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
CA
CM CE
2) Do 3 and TCM form a linear pair? Justify your answer.
No. Their noncommon sides are not opposite rays.
When two lines intersect, ____ angles are formed.four
12
34
There are two pair of nonadjacent angles.
These pairs are called _____________.vertical angles
Vertical AnglesVertical Angles
Definition of
VerticalAngles
Two angles are vertical iff they are two nonadjacent angles formed by a pair of intersecting lines.
12
34
Vertical angles:
1 and 3
2 and 4
Vertical AnglesVertical Angles
Theorem 3-1
Vertical AngleTheorem
Vertical angles are congruent.
1
4
3
2mn
1 3
2 4
Vertical AnglesVertical Angles
Find the value of x in the figure:
The angles are vertical angles.
So, the value of x is 130°.130°
x°
Vertical AnglesVertical Angles
Find the value of x in the figure:
The angles are vertical angles.
(x – 10) = 125.(x – 10)°
125°
x – 10 = 125.
x = 135.
Vertical AnglesVertical Angles
Definition of
Complementary
Angles
30°
A
BC
60°D
E
F
Two angles are complementary if and only if (iff) The sum of their degree measure is 90.
mABC + mDEF = 30 + 60 = 90
Complementary and Supplementary AnglesComplementary and Supplementary Angles
30°
A
BC
60°D
E
F
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.
Complementary angles DO NOT need to have a common side or even the same vertex.
Complementary and Supplementary AnglesComplementary and Supplementary Angles
15°H
75° I
Some examples of complementary angles are shown below.
mH + mI = 90
mPHQ + mQHS = 9050°
H
40°Q
P
S
30°60°T
UV
WZ
mTZU + mVZW = 90
Complementary and Supplementary AnglesComplementary and Supplementary Angles
Definition of
Supplementary
Angles
If the sum of the measure of two angles is 180, they form a special pair of angles called supplementary angles.
Two angles are supplementary if and only if (iff) the sum of their degree measure is 180.
50°
AB
C
130°
D
E F
mABC + mDEF = 50 + 130 = 180
Complementary and Supplementary AnglesComplementary and Supplementary Angles
105°H
75° I
Some examples of supplementary angles are shown below.
mH + mI = 180
mPHQ + mQHS = 18050°
H
130°
Q
P S
mTZU + mUZV = 180
60°120°
T
UV
W
Z
60° and
mTZU + mVZW = 180
Complementary and Supplementary AnglesComplementary and Supplementary Angles
Suppose A B and mA = 52.
Find the measure of an angle that is supplementary to B.
A52°
B52° 1
B + 1 = 1801 = 180 – B1 = 180 – 521 = 128°
Congruent AnglesCongruent Angles
1) If m1 = 2x + 3 and the m3 = 3x + 2, then find the m3
2) If mABD = 4x + 5 and the mDBC = 2x + 1, then find the mEBC
3) If m1 = 4x - 13 and the m3 = 2x + 19, then find the m4
4) If mEBG = 7x + 11 and the mEBH = 2x + 7, then find the m1
x = 17; 3 = 37°
x = 29; EBC = 121°
x = 16; 4 = 39°
x = 18; 1 = 43°
A B C
D
E
G
H
12
34
Congruent AnglesCongruent Angles