1Geometry Lesson: Rays, Angles

Aim:Do Now:1) Given points A(8, -5) and B(0, -11):

a) Determine the coordinates of the midpoint

of AB. b) Determine AB.

2) Sketch angles with the following measures: a) Less than 90 b) Exactly 90 c) Greater than 90 d) Exactly 180

Ans. M(4, -8)Ans. 10

What are rays and angles?

2Geometry Lesson: Rays, Angles

Def: Rays

PA

A ray is a part of a line that consists of an endpoint, and all points on one side of the endpoint.

Def: Opposite RaysOpposite Rays are two rays of the same line with a common endpoint and no other points in common.

A

BP

PA�������������� and PB�������������� are opposite rays

PA��������������

= “Ray PA”

3Geometry Lesson: Rays, Angles

Def:Angle: An angle is the union of two rays having the same endpoint.

side

sidevertex

Three capital letters, with vertex in the middle:

Single lowercase letter or number inside the angle:

B

A

C

x

x

Naming Angles:a)

b)

c) Use the name of the vertex angle if it’s the only angle at that vertex: B

ABC or CBA

4Geometry Lesson: Rays, Angles

Angle Measure:

B

A

C25°

Def: Congruent Angles are angles having equal measure.

If ABC EFG , then m ABC m EFG

A B C Ans.

The measure of an angle is the number of degrees in the angle.

25°m ABC

25°m ABC

Q: Which of the following angles are congruent?A)

45°A

C) C

45°

B)

B

45°

5Geometry Lesson: Rays, Angles

Def: Straight Angle

A BO

A straight angle is the union of two opposite rays.

Straight angles have a measure of 180°

180°m AOB

Def: Right Angle:

M Q

P

A right angle has a measure of 90°

90°m PMQ

6Geometry Lesson: Rays, Angles

Def: Acute Angle:

Y

X

Z

An acute angle has a measure greater than 0° and less than 90 °.

0° 90°m XYZ

Def: Obtuse Angle:

Q

X R

An obtuse angle has a measure greater than 90° and less than 180°.

90° 180°m QXR

7Geometry Lesson: Rays, Angles

Def: Perpendicular Lines are two lines that intersect to form

right angles.

l

m m l

H K

L

J

HK JL Right Angles: HJL KJLStraight Angle: HJK

Are lines m and l perpendicular?

NOT UNLESS SPECIFIED BY THE GIVEN INFO OR A BOX IN THE DIAGRAM !!!!

8Geometry Lesson: Rays, Angles

Addition/Subtraction of Angles: If several angles share a common

vertex, we can write addition and subtraction expressions using the names of the angles.

R

T

S

20°

38°

P

? TSP PSR TSRm TSP m PSR m TSR

20° + 38° = 58° 58° - 38° = 20°

? TSR PSR TSPm TSR m PSR m TSP

9Geometry Lesson: Rays, Angles

Def: Angle Bisector:

How do we bisect EFG ?

Ans.: Make a ray from the vertex that divides it in half.

80°

G

E

F

40°

40°

G

E

F

P

The bisector of an angle is a ray that divides the angle into two congruent angles.

Conclusions: If FP��������������

bisects EFG , then: A) EFP PFG Or m EFP m PFG

B)1

2m EFP m EFG C)

1

2m PFG m EFG

10Geometry Lesson: Rays, Angles

Angle Bisector ExamplesEx #1

PT

M

N

R

S

bisects PT NPR��������������

a) State a pair of congruent angles.

NPT TPR

c) If 9 5m NPT x and 6 7m TPR x , find x.

x = 4

b) Is ?MPT TPS There is no way to tell from the given info.

11Geometry Lesson: Rays, Angles

Ex# 2 Simple 2-Column Proof: A

B

C

D

Statement Reason 1) 1) 2) 2) 3) 3)

bisects DB ADC��������������

ADB BDC Given Def. of Angle Bisector Def. of Congruent Angles

G i v e n : b i s e c t s D B A D C��������������

P r o v e : m A D B m B D C

m ADB m BDC