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1.3 a: Angles, Rays, Angle Addition, Angle Relationships
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.
CCSS
Rays
• A ray extends forever in one direction
• Has one endpoint
• The endpoint is used first when naming the ray
R
B ray RB
T
Wray WT
R
B
R
B
R
B
R
Angles
• Angles are formed by 2 non-collinear rays
• The sides of the angle are the two rays
• The vertex is where the two rays meet
ray ray
Vertex- where they met
Angles (cont.)
• Measured in degrees
• Congruent angles have the same measure
Naming an Angle
You can name an angle by specifying three points: two on the rays and one at the vertex.
• The angle below may be specified as angle ABC or ABC. The vertex point is always given in the middle. Named:
1) Angle ABC2) Angle CBA3) Angle B * *you can only use the vertex if there is ONE angle
Vertex
Ex. of naming an angle
• Name the vertex and sides of 4, and give all possible names for 4.
4 5
W X Z
T
Vertex:
Sides:
Names:
X
XW & XT
WXT
TXW
4
Name the angle shown as
Angles can be classified by their measures
• Right Angles – 90 degrees
• Acute Angles – less than 90 degrees
• Obtuse Angles – more than 90, less than 180
Angle Addition Postulate
• If R is in the interior of PQS, then
m PQR + m RQS = m PQS.
P
Q
S
R30 20
Find the m< CAB
Example of Angle Addition Postulate
Ans: x+40 + 3x-20 = 8x-60
4x + 20 = 8x – 60
80 = 4x
20 = x
Angle PRQ = 20+40 = 60
Angle QRS = 3(20) -20 = 40
Angle PRS = 8 (20)-60 = 100
60 40
100
4a+9
4a+9
-2a+48
Find the m< BYZ
Types of Angle Relationships
1. Adjacent Angles
2. Vertical Angles
3. Linear Pairs
4. Supplementary Angles
5. Complementary Angles
1) Adjacent Angles
• Adjacent Angles - Angles sharing one side that do not overlap
1
2
3
2)Vertical Angles
• Vertical Angles - 2 non-adjacent angles formed by 2 intersecting lines (across from each
other). They are CONGRUENT !!
1 2
3) Linear Pair
• Linear Pairs – adjacent angles that form a straight line. Create a 180o angle/straight angle.
1
2
3
4) Supplementary Angles
• Supplementary Angles – two angles that add up to 180o (the sum of the 2 angles is 180)
Are they different from linear pairs?
5) Complementary Angles
• Complementary Angles – the sum of the 2 angles is 90o
Angle BisectorAngle Bisector
• A ray that divides an angle into 2 congruent adjacent angles.
BD is an angle bisector of <ABC.
B
A
C
D
YB bisects <XYZ
40
What is the m<BYZ ?
Last example: Solve for x.
x+40o
3x-20o
x+40=3x-20
40=2x-20
60=2x
30=x
B C
D
BD bisects ABC
A
Why wouldn’t the Angle Addition Postulate help us solve this initially?
Solve for x and find the m<1
Solve for x and find the m<1
Find x and the DBC