1.3 a: Angles, Rays, Angle Addition,
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1.3 a: Angles, Rays, Angle Addition, Angle Relationships. CCSS.

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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

B

B

B

ray RB

R

R

R

R

R

T

ray WT

W


Angles
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
Angles (cont.)

  • Measured in degrees

  • Congruent angles have the same measure


Naming an angle
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:

  • Angle ABC

  • Angle CBA

  • Angle B * *you can only use the

  • vertex if there is ONE

  • angle

Vertex


Ex of naming an angle
Ex. of naming an angle

  • Name the vertex and sides of 4, and give all possible names for 4.

T

Vertex:

Sides:

Names:

X

XW & XT

WXT

TXW

4

4 5

W X Z



Angles can be classified by their measures
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
Angle Addition Postulate

  • If R is in the interior of PQS, then

    m PQR + m RQS = m PQS.

P

R

30

20

Q

S



Example of angle addition postulate
Example of Angle Addition Postulate

100

Ans: x+40 + 3x-20 = 8x-60

4x + 20 = 8x – 60

80 = 4x

20 = x

40

60

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
Types of Angle Relationships

  • Adjacent Angles

  • Vertical Angles

  • Linear Pairs

  • Supplementary Angles

  • Complementary Angles


1 adjacent angles
1) Adjacent Angles

  • Adjacent Angles - Angles sharing one side that do not overlap

2

1

3


2 vertical angles
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
3) Linear Pair

  • Linear Pairs – adjacent angles that form a straight line. Create a 180o angle/straight angle.

2

1

3


4 supplementary angles
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
5) Complementary Angles

  • Complementary Angles – the sum of the 2 angles is 90o


Angle bisector
Angle Bisector

  • A ray that divides an angle into 2 congruent adjacent angles.

    BD is an angle bisector of <ABC.

A

D

B

C


YB bisects <XYZ

40

What is the m<BYZ ?


Last example solve for x
Last example: Solve for x.

BD bisects ABC

A

D

x+40o

x+40=3x-20

40=2x-20

60=2x

30=x

3x-20o

C

B

Why wouldn’t the Angle Addition Postulate help us solve this initially?





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