Unit 1 Describe and Identify the three undefined terms, Understand Segment Relationships and Angle...

Unit 1

Describe and Identify the three undefined terms, Understand Segment Relationships and Angle Relationships

Part 1

Definitions:

Points, Lines, Planes and Segments

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:

1. Name two points.2. Name two lines.3. Name two segments.4. Name two rays.

5. Name a line that does not contain point T.6. Name a ray with point R as the endpoint.7. Name a segment with points T and Q as its endpoints.8. Name three collinear points.9. Name three non-collinear points.

Congruent

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

Symbol:

Example:

≅

AB ≅ CD

< >

>< A B

C D

Midpoint / Segment Bisector

The midpoint of a segment is the point that divides the segment into two congruent segments

The Segment Bisector is a segment, line or ray that intersects another segment at its midpoint.

Example

Q is the Midpoint of PR, if PQ=6x-7 and QR=5x+1, find x, PQ, QR, and PR.

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

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

Example

Find the length XY in the figure shown.

Example

If S is between R and T and RS = 8y+4, ST = 4y+8, and RT = 15y – 9. Find y.

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 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

P

Q

S

R

Example

If m<BAC = 155, find m<CAT and m<BAT

(3x+14)°

(4x-20)°B

A

TC

Example

<ABC is a straight angle, find x.

(2x+10)°(11x-12) °A

C

D

B

Angle Bisector

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

Example

Ray KM bisects <JKL, if m<JKL=72 what is the m<JKM?

Adjacent Angles

are two angles that lie in the same plane, have a common vertex, and a common side, but no common interior points

∠ADB is adjacent to ∠BDC

AD

C

B

Vertical Angles

Two non-adjacent angles formed by two intersecting lines

Vertical Angles have the same measure and are congruent

∠1 is vertical to ∠2

21

Linear Pair

A pair of adjacent angles who are also supplementary

∠1 and ∠2 are a liner pairand m∠1 + m∠2 =180

21

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

Examples

Part 3

Polygons

Polygon

Closed figure whose sides are all segments.– To be a Polygon 2 things must be true

• Sides have common endpoints and are not collinear

• Sides intersect exactly two other sidesNon-ExamplesExamples

Naming a Polygon

The sides of each angle in a polygon are the sides of the polygon

The vertex of each angle is a vertex of the polygon

They are named using all the vertices in consecutive

order Example

A

D

C

B

The number of sides determines the name of the polygon

3 - Triangle 4 - Quadrilateral5 - Pentagon6 - Hexagon7 - Heptagon8 - Octagon9 - Nonagon10 - Decagon12 - DodecagonAnything else …. N - gon (where n represents the number of sides)

Concave VS Convex

ConvexConcave

Regular Polygon

A regular polygon is a convex polygon whose sides are all congruent and whose angles are all congruent

Perimeter

The perimeter of a polygon is the sum of the lengths of its sides.

p = l + w + l + w

p = 2l + 2w

p = s + s + s + s p = 4s

p = a + b + c

ww

l

l

s

s

s

s

c

ba

Example

Perimeter of the Coordinate Plane

Find the perimeter of the triangle ABC with A(-5,1), B(-1,4), C(-6,-8)

Area

Area of a polygon is the number of square units it encloses

A = bhA =

1

2bh

h

b

h

b

Circle

C = 2πr

A = πr2

r

Unit 1

The End!