Points, Lines, and Planes

The Three Undefined Terms of Geometry

A point is an exact location in space.

You are here.

A true point has no length, no width, and no height.

In fact, you cannot see a true point.

A point is named by a letter.

P

Point P

Lines are 1-dimensional objects that have only length. Lines

continue forever in both directions.

Because a line has no width or height, you cannot see a true line.

A line is defined by 2 points.

A

BLine AB

A plane

is a flat

surface that has

length & width

but no height.

You can see a plane only if you view it at a certain angle.

A true plane goes on forever in all directions.

A true plane goes on forever in all directions.

A true plane goes on forever in all directions.

A true plane goes on forever in all directions.

A true plane goes on forever in all directions.

Planes can…

Planes can intersect.

An X-Wing fighter from Star Wars has wings that intersect.

Planes can be perpendicular.

Planes can be parallel.

A Tie Fighter from Star Wars has wings that are parallel.

Planes can be…intersecting

perpendicular

parallel

A ray has a starting point but no ending point.

A ray of light has a starting point (like the sun) and continues

forever in the same direction.

Of course, if the ray hits an object, the light could be

absorbed or reflected.

MIRROR

A ray is also defined by 2 points.

C

DRay CD

26Geometry 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”

A line segment has a starting point and an ending point. Line

segments can be measured.

A line segment also defined by 2 points.

E

F Line Segment EF

Lines can do various things.

Lines can intersect at a point to form angles.

X

Y

Z

Angles are defined by 3 points.

X

Z

Y

XYZ

32Geometry 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

These angles can be right angles.

90

When lines intersect to form right angles, they are said to be perpendicular.

V

R

TU

cS

UT RV

These angles can be right angles.

c

R

S TU

V

m RST = 90⁰m USR = 90⁰

m USV = 90⁰ m TSV = 90⁰

Lines can also run into each other to form straight angles.

37Geometry 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

180 is a straight angle

Lines can intersect to form acute and obtuse angles.

135

obtuse

45 acute

40Geometry 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

41Geometry 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°

42Geometry 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 !!!!

Lines do not always intersect.

Lines can be parallel.

Parallel lines have the same slope or steepness.

Is it possible for lines not to intersect and not be parallel

either?

Believe it or not, this is possible.Let’s consider a 3-dimensional

rectangular prism.

These 2 lines are not parallel, but they are not intersecting either.

These lines are called skew lines.

You might have heard the word “skewer” the last time you had a

barbecue.

                     

   

S

K

E

W

E

R

A skewer raises your “dinner” off the surface of the grill. The skewer does not intersect the grill, and the skewer is not

parallel to the grill.

Lines can be…

intersecting

perpendicular

90º

parallelskew

The end.