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