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# 1.3 a: Angles, Rays, Angle Addition, Angle Relationships - PowerPoint PPT Presentation

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

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Presentation Transcript

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.

• 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 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

• Measured in degrees

• Congruent angles have the same measure

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

• 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

• Right Angles – 90 degrees

• Acute Angles – less than 90 degrees

• Obtuse Angles – more than 90, less than 180

• If R is in the interior of PQS, then

m PQR + m RQS = m PQS.

P

R

30

20

Q

S

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

-2a+48

4a+9

4a+9

• Vertical Angles

• Linear Pairs

• Supplementary Angles

• Complementary Angles

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

2

1

3

• Vertical Angles - 2 non-adjacent angles formed by 2 intersecting lines (across from each other). They are CONGRUENT !!

1

2

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

2

1

3

• Supplementary Angles – two angles that add up to 180o (the sum of the 2 angles is 180)

Are they different from linear pairs?

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

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

BD is an angle bisector of <ABC.

A

D

B

C

40

What is the m<BYZ ?

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?