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 LetterNo size or shape.Verbally you say “Point P”

P

Line

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”

AB

l

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, you’ll need only one number.

ABl

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.

R

F

K

Plane RFKPPlane 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 you’ll 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.

AB

AB

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.

AC

B

Notice that B is between but not in exact middle.

Example

L M NFind the length of LNor 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 = 3y+1?

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.

AC FGHint: Shapes can be congruent, measurements can only be equal. So if you’re talking about a shape, you say congruent or not congruent!

Congruence

Congruence can not be assumed!Don’t 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.

A C

P

Q

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)

1 2 3 4 5 6

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 ± 1/2So the segment could be as small as 3 ½” or as

big as 4 ½” and still be called 4”.

Precision (Con’t)

1 2 3 4 5 6

Here we have the same segment but a different, more accurate measuring device.

The units are broken down into ¼’s. 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.

MidpointDefinition - 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.

AC

A B C

One Dimensional

-3 -2 -1 0 1 2 3 4

If point A was at -3 and point B was at 2, then AB=5 b/c the formula for AB = |A – B|

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

CD = |4 – (-2)| or | -2 – 4| = 6

The MP formula is (A+B)/2 (-3+2)/2 = -1/2

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

A BC D

Two Dimensional

2 22 1 2 1( )d x x y y

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.

1 2 1 2,2 2

x x y yMP

Distance (Shortcut)

(2, 5)

(8, 10)

Find the distance between these two points.

d=√(8 – 2)2 + (10 – 5)2 = √36 + 25 = √61

Or use the Pythagorean theorem.

Create a right triangle.

6

5

d2 = 62 + 52 = 36 + 25 = 61 so d = √61

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.

ZXRRRRRRRRRRRRRR

ZX

XZ

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.

A

C

BBA

BC

RRRRRRRRRRRRRR

RRRRRRRRRRRRRR

Angles Angles are created by two non-

collinear rays that share a common end point.

C

DE

Angles are named by using one letter from one side, the vertex angle, and one letter from the other side.

<CED or <DEC

An angle consists of two sides which are rays and a vertex which is a point.

C

DE

InteriorExterior

ExteriorExterior

Interior vs. Exterior

Classifications of Angles

Right Angle – An angle with a measurement of exactly 90° m<ABC=90°Acute Angle – An angle with a measurement more than 0° but less than 90° 0° < m<ABC < 90°Obtuse Angle – An angle with a measurement more than 90° but less than 180° 90° < m<ABC < 180°

Congruence of Angles

Angles with the same measurement are said to be congruent.

C

25°25°

A D

E G

m<ACE = 25° and m<DCG = 25°… since the two angles have the same measurement we can say that they’re congruent. ACE DCG

Angle Bisector

An angle bisector is a Ray that divides an angle into two congruent angles.

A

D

H

P

If is an angle bisector….Then <ADP is congruent to <PDH.

DPRRRRRRRRRRRRRR

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.

A

B

C

D

<ABC and <CBD are Adjacent Angles. They don’t have to be equal.

Common Vertex? Common Side?B BCRRRRRRRRRRRRRR

No Common Interior Point?

Vertical Angles Vertical Angles – Are two non-adjacent

angles formed by intersecting lines.A

B

C

DE

Two Intersecting Lines?

<ABD and <CBE are non-adjacent angles formed by intersecting lines. They are Vertical Pair.

What else?<ABC and <DBE are also Vertical Pair.

Linear Pair

Linear Pair – Is a pair of adjacent angles whose non-common sides are opposite rays.

L M N

P

Are <LMP and PMN are Adjacent? Yes! Are Ray’s ML and MN the Non-

Common Sides?Yes!

Are Ray’s ML and MN Opposite Rays? Yes!

<LMP and <PMN 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!

12

<1 and <2 are Comp.

25°

65°

1

2

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 and <2 are Supp.

25°

155°

1

2L M N

P1 2

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 ΔABC.

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 – Quadrilateral5 – Pentagon 6 – Hexagon7 – Heptagon 8 – Octagon9 – Nonagon 10 – Decagon11 – 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 TriangleQuadrilateral - 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.