Transcript 1.3a angles rays angle addition (1)

1.3 a: Angles, Rays, Angle Addition,
Angle Relationships
CCSS
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.
Rays
• A ray extends forever in one direction
• Has one endpoint
• The endpoint is used first when naming
the ray
B
ray RB
R
T
ray WT
W
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
Vertex- where they met
ray
ray
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 ABC
2) Angle CBA
3) 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.
T
Vertex: X
4
W
X
5
Sides:
Z
Names:
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
R
Q
30
20
S
Find the m< CAB
Example of Angle Addition
Postulate
100
Ans: x+40 + 3x-20 = 8x-60
60
40
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
Find the m< BYZ
-2a+48
4a+9
4a+9
Types of Angle Relationships
1.
2.
3.
4.
5.
Adjacent Angles
Vertical Angles
Linear Pairs
Supplementary Angles
Complementary Angles
1) Adjacent Angles
• Adjacent Angles - Angles sharing one side
that do not overlap
2
1
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.
2
1
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 Bisector
• A ray that divides an angle into 2 congruent
adjacent angles.
A
D
B
C
BD is an angle
bisector of <ABC.
YB bisects <XYZ
40
What is the m<BYZ ?
Last example: Solve for x.
BD bisects ABC
A
D
x+40=3x-20
40=2x-20
B
C
60=2x
30=x
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