unit 1 learning outcomes 1: describe and identify the three undefined terms learning outcomes 2:...

Unit 1

Learning Outcomes 1: Describe and Identify the three undefined terms

Learning Outcomes 2: Understand Angle Relationships

Part 1

Definitions:

Points, Lines and Planes

Undefined Terms

Points, Line and Plane are all considered to be undefined terms.– This is because they can only be explained

using examples and descriptions.– They can however be used to define other

geometric terms and properties

Point– A location, has no shape or size– Label:

Line– A line is made up of infinite points and has no thickness or width, it will continue

infinitely.There is exactly one line through two points.– Label:

Line Segment– Part of a line– Label:

Ray– A one sided line that starts at a specific point and will continue on forever in one

direction.– Label:

< >A B

Collinear – Points that lie on the same line are said to be

collinear – Example:

Non-collinear– Points that are not on the same line are said to be

non-collinear (must be three points … why?)– Example:

< >

F

A BE

Plane– A flat surface made up of points, it has no depth

and extends infinitely in all directions. There is exactly one plane through any three non-collinear points

Coplanar– Points that lie on the same plane are said to be

coplanar

Non-Coplanar– Points that do not lie on the same plane are said to

be non-coplanar

Intersect

The intersection of two things is the place they overlap when they cross. – When two lines intersect they create a

point.– When two planes intersect they create a

line.

Space

Space is boundless, three-dimensional set of all points. Space can contain lines and planes.

Practice Use the figure to give examples of the following:

Name two points.Name two lines.Name two segments.Name two rays.

Name a line that does not contain point T.Name a ray with point R as the endpoint.Name a segment with points T and Q as its endpoints.Name three collinear points.Name three non-collinear points.

QuickTime™ and a decompressor

are needed to see this picture.

Part 2

Distance, Midpoint and Segments

Distance Between Two Points

Distance on a number line • PQ = or

Distance on coordinate plane – The distance d between two points with

coordinates is given by

B−A A−B

x1, y1( )and x2 ,y2( )

d = x2 −x1( )2+ y2 −y1( )

2

Examples

Example 1:– Find the distance between (1,5) and (-2,1)

Examples 2: – Find the distance between Point F and

Point B

-1-6< >

BE

Congruent

When two segments have the same measure they are said to be congruent

Symbol:

Example:

≅

AB ≅ CD

< >

>< A B

C D

Between

Point B is between point A and C if and only if A, B and C are collinear and

AB + BC =AC

< >A B C

Midpoint

Midpoint– Halfway between the endpoints of the

segment. If X is the MP of then AB

AX =XB

< >XA B

Finding The Midpoint

Number Line– The coordinates of the midpoint of a segment

whose endpoints have coordinates a and b is

Coordinate Plane– The coordinates of midpoint of a segment whose

endpoints have coordinates

are

a +b2

x1, y1( )and x2 ,y2( )x1 + x2

2,y1 + y2

2⎛⎝⎜

⎞⎠⎟

Examples

The coordinates on a number line of J and K are -12 and 16, respectively. Find the coordinate of the midpoint of

Find the coordinate of the midpoint of

for G(8,-6) and H(-14,12).

Segment Bisector

A segment bisector is a segment, line or plane that intersects a segment at its midpoint.

Segment Addition Postulate

– if B is between A and C, then

AB + BC = AC

– If AB + BC = AC, then B is between

A and C

Part 3

Angles

Angle

An angle is formed by two non-collinear rays that have a common endpoint. The rays are called sides of the angle, the common endpoint is the vertex.

Kinds of angles

Right Angle

Acute Angle

Obtuse Angle

Straight Angle / Opposite Rays

Congruent Angles

Just like segments that have the same measure are congruent, so are angles that have the same measure.

Angle Bisector

A ray that divides an angle into two congruent angles is called an angle bisector.

Angle Addition Postulate

– If R is in the interior of <PQS, then m<PQR + m<RQS = m<PQS

– If m<PQR + m<RQS = m<PQS, then R is in the interior of <PQS

Measuring Angles

How to use a protractor. – 1.) Line up the base line with one ray of

your angle. – 2.) Follow the base line out to zero, if you

are at 180 switch the protractor around.– 3.) Trace to protractor up until you reach

the second ray of your angle.– 4) The number your finger rests on is your

angle measure.

Part 4

Angle Relationships

Pairs of 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 Vertical Angles-are two non-adjacent angles formed by two intersecting linesLinear Pair - is a pair of adjacent angles who are also supplementary

Angle Relationships

Complementary Angles - Two angles whose measures have a sum of 90

Supplementary Angles - are two angles whose measures have a sum of 180

Part 5

Angle Theorems

Theorem 2.3

Supplement Theorem - – If two angles form a linear pair, then they

are supplementary angles

Theorem 2.4

Complement Theorem– If the non-common sides of two adjacent

angles form a right angle, then the angles are complementary angles.

Theorem 2.6

Angles supplementary to the same angle or to congruent angles are congruent

Theorem 2.7

Angles complementary to the same angle or to congruent angles are congruent

Theorem 2.8

Vertical Angles Theorem– If two angles are vertical, then they are

congruent

Part 6

Perpendicular Lines and their theorems

Perpendicular Lines

Lines that form right angles are perpendicular– Perpendicular lines intersect to form 4 right angles– Perpendicular lines form congruent adjacent

angles– Segments and rays can be perpendicular to lines

or to other line segments or rays– The right angle symbol in a figure indicates that

the lines are perpendicular.

Theorems

Theorem 2.9 - Perpendicular lines intersect to form four right angles

Theorem 2.10 - All right angles are congruent

Theorem 2.11 - Perpendicular lines form congruent adjacent angles

More Theorems

Theorem 2.12 - If two angles are congruent and supplementary, the each angle is a right angle

Theorem 2.13 - If two congruent angles form a linear pair, then they are right angles.

Unit 1

The End!