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1.1 Points, Lines and Planes

Undefined Terms

- There are three undefined terms in Geometry.
- They are Points, Lines and Planes.
- They are considered undefined because they have

only been explained using examples and

descriptions.

Points

- Points are simply locations.
- Drawn as a dot.
- Named by using a Capital Letter
- No size or shape.
- Verbally you say Point P

P

Line

l

- A line is a collection of an infinite number of

points (named or un-named). - Points that lie on the line are called Collinear.
- Collinear Points are points that are on the same

line. - Draw a line with arrows on each end to signify

that it is infinite in both directions. - Name by either two points on the line or lower

case script letter

Line (Continued)

- A line has only one dimension (length).
- It has no width or depth.
- Postulate There exists exactly one line through

two points. - To plot a point on a number line, youll need

only one number.

l

Plane

- A plane is a flat surface made up of an infinite

number of points. - Points that lie on the same plane are said to be

Coplanar. - Planes are named by using a capital, script

letter or three non-collinear points.

Plane RFK

Plane P

P

Plane (Continued)

- Although a plane looks like it is a

quadrilateral, it is in fact infinitely long and

wide. - Planes (Coordinate Plane) have two dimensions

so you need two numbers to plot a point. P(x,y)

Space

- Space is a boundless, three dimensional set of

all points. Space can contain, points, lines and

planes. - In chapter 13 you will see that youll need three

numbers to plot a point in space. P(x,y,z)

Describing What you see!

- There are key terms such as
- Lies in,
- Contains,
- Passes through,
- Intersection,
- See Pg 12.

1.2 Linear Measure and Precision

Introduction

- Lines are infinitely long.
- There are portions of lines that are finite. In

other words, they have a length. - The portion of a line that is finite is called a

Line Segment. - A line segment or segment has two distinct end

points.

Betweenness

- Betweenness of points is the relationships among

three collinear points. - We can say B is between A and C and you should

think of this picture.

C

A

B

Notice that B is between but not in exact middle.

Example

Find the length of LN or LN?

From this picture we can always write this

equation LM MN LN.

So, if LM 3 and MN 5, we can say that LN 8.

What if LM 2y, MN 21 and LN 3y1?

Then we can write.. 2y 21 3y 1

From this equation we can solve for y and

substitute that value to find LN.

Congruence of Segments

- Segments can be Congruent if they have the same

measurement. - We have a special symbol for congruent. It is an

equal sign with a squiggly line above it.

Hint Shapes can be congruent, measurements can

only be equal. So if youre talking about a

shape, you say congruent or not congruent!

Congruence

- Congruence can not be assumed!
- Dont think, that just because it looks like the

same length, it is. - Short cut we can use congruent marks to show

that segments are congruent.

C

Q

A

P

Precision (H)

- The precision of a measurement depends on the

smallest unit of measure available on the

measuring tool. - The precision will always be ½ the smallest unit

of measure of the measuring device.

Precision (H)

- Here to find the length we would have to say it

is four units long b/c it is closer to 4 than 5.

- The precision of this measuring device is ½ the

smallest unit of measure, 1, or the precision is

1/2.

- We can say the measurement is 4 1/2

- So the segment could be as small as 3 ½ or as

big as 4 ½ and still be called 4.

Precision (Cont)

- Here we have the same segment but a different,

more accurate measuring device.

- The units are broken down into ¼s. The segment

is closer to 4 ¼ than 4 ½.

- The precision is ½ of ¼, or 1/8th.

- So the length is 4 ¼ 1/8th.

Smallest 4 1/8th Largest 4 3/8th.

1.3 Distance and Midpoints

Distance

- The coordinates of the two endpoints of a line

segment can be used to find the length of the

segment. - The length from A to B is the same as it is from

B to A. - Thus AB BA (This stands for the measurement of

the segment) - Distance (length) can never be negative.

Midpoint

- Definition - The midpoint of a segment is the

point ½ way between the endpoints of the segment. - If B is the Midpoint (MP) of
- then, AB BC.
- The midpoint is a location, so it can be positive

or negative depending on where it is.

One Dimensional

B

C

D

A

-3 -2 -1 0 1 2 3 4

- If point A was at -3 and point B was at 2, then

AB5 b/c the formula for AB A B

- The MP formula is (AB)/2

(-32)/2 -1/2

- What if point C was at -2 and D was at 4,
- what is CD?

CD 4 (-2) or -2 4 6

MP is (4 (-2))/2 1

Two Dimensional

- We designate points on a plane using ordered

pair P(x,y). - We plot them on the Cartesian Coordinate plane

just as you did in Alg I. - Again, distances can not be negative because

lengths are not negative. - Midpoints can be either positive or negative b/c

it is simply a location.

Distance (Shortcut)

(8, 10)

5

Find the distance between these two points.

6

(2, 5)

Or use the Pythagorean theorem.

Create a right triangle.

d2 62 52 36 25 61 so d v61

1.4 Angle Measure

Another Portion of a Line

- We already talked about segments, now let us talk

about Rays. - A ray is a portion of a line that has only one

end point. It is infinite in the other

direction. - A ray is named by using the end point and any

other point on the ray.

Opposite Rays

- If you chose a point on a line, that point

determines exactly two rays called Opposite Rays. - These two opposite rays form a line and are said

to be collinear rays.

C

A

B

Angles

- Angles are created by two non-collinear rays

that share a common end point.

ltCED or ltDEC

- Angles are named by using one letter from one

side, the vertex angle, and one letter from the

other side.

- An angle consists of two sides which are rays

and a vertex which is a point.

Interior vs. Exterior

Exterior

Interior

Exterior

Exterior

Classifications of Angles

- Right Angle An angle with a measurement of

exactly 90 mltABC90 - Acute Angle An angle with a measurement more

than 0 but less than 90 0 lt mltABC lt 90 - Obtuse Angle An angle with a measurement more

than 90 but less than 180 90 lt mltABC lt 180

Congruence of Angles

- Angles with the same measurement are said to be

congruent.

- mltACE 25 and mltDCG 25 since the two

angles have the same measurement we can say that

theyre congruent.

Angle Bisector

- An angle bisector is a Ray that divides an angle

into two congruent angles.

P

- If is an angle bisector.
- Then ltADP is congruent to ltPDH.

1.5 Angle Relationships

- Angle Pairs

Adjacent Angles

- Adjacent Angles Are two angles that lie in the

same plane, have a common vertex, and a common

side but no common interior points.

ltABC and ltCBD are Adjacent Angles. They dont

have to be equal.

Common Side?

Common Vertex?

B

No Common Interior Point?

Vertical Angles

- Vertical Angles Are two non-adjacent angles

formed by intersecting lines.

- Two Intersecting Lines?

ltABD and ltCBE are non-adjacent angles formed by

intersecting lines. They are Vertical Pair.

What else?

ltABC and ltDBE are also Vertical Pair.

Linear Pair

- Linear Pair Is a pair of adjacent angles whose

non-common sides are opposite rays.

- Are ltLMP and PMN are Adjacent?

Yes!

- Are Rays ML and MN the Non-Common Sides?

Yes!

- Are Rays ML and MN Opposite Rays?

Yes!

ltLMP and ltPMN are Linear Pair!

Complementary Angles

- Complementary Angles Are two angles whose

measures have a sum of 90 - Do you see the word Adjacent in the definition?

No!

1

2

lt1 and lt2 are Comp.

Supplementary Angles

- Supplementary Angles Are two angles whose

measures have a sum of 180 - Do you see the word Adjacent in the definition?

No!

1

2

lt1 and lt2 are Supp.

Perpendicular Lines

- Perpendicular Lines intersect to form four right

angles. - Perpendicular Lines intersect to form congruent,

adjacent angles. - Segments and rays can be perpendicular to lines

or to other line segments or rays. - The right angle symbol indicates that the lines

are perpendicular.

Assumptions

- Things that can be assumed.
- Coplanar, Intersections, Collinear, Adjacent,

Linear Pair and Supplementary - Things that can not be assumed.
- Congruence, Parallel, Perpendicular, Equal, Not

Equal, Comparison.

1.6 Polygons

Polygon

- Polygon A closed figure whose sides are all

segments and they only intersect at the end

points of the segments. - Polygons are named by using consecutive points at

the vertices. - Example A triangle with points of A, B and C is

named ?ABC.

Concave vs. Convex

- Concave A polygon is concave when at least one

line that contains one of the sides passes

through the interior. - Convex A polygon is convex when none of the

lines that contains sides passes through the

interior.

Concave

Convex

Classification by Sides

- Polygons are classified by the number of sides it

has. - 3 Triangle 4 Quadrilateral
- 5 Pentagon 6 Hexagon
- 7 Heptagon 8 Octagon
- 9 Nonagon 10 Decagon
- 11 Undecagon 12 Dodecagon
- Any polygon more than 12 then N-Gon. Example

24 sides is a 24-gon.

Regular Polygon

- Regular Polygon Is a polygon that is

equilateral (all sides the same length),

equiangular (all angles the same measurement) and

convex. - Examples
- Triangles Equilateral Triangle
- Quadrilateral - Square

Perimeter

- Perimeter The sum of the lengths of all the

sides of the polygon. - May have to do distance formula for coordinate

geometry problem. - See example 3.