**Section 1.2**

Points, Lines, and Planes

Points, Lines, and Planes

**Point -**

indicates a location and has no size

Points are usually represented by a single dot and named with a single capital letter

indicates a location and has no size

Points are usually represented by a single dot and named with a single capital letter

Section 1.3

Measuring Segments

Postulate 1-5 (Ruler Postulate)

Every point on a line can be paired with a real number. This makes a one-to-one correspondence between the points on the line and the real numbers.

The real number that corresponds to the point is called the

coordinate

of the point.

Section 1.4

Measuring Angles

Angle -

formed by two rays with the same endpoint.

The rays are the

sides

of the angle. The endpoint is the

vertex

of the angle.

You can name an angle in three ways:

1. By its vertex

2. A point on each ray and the vertex (vertex

must

be the letter in the middle)

3. A number

Section 1.5

Exploring Angle Pairs

Adjacent Angles -

two coplanar angles with a common side, a common vertex, and no common interior points

Vertical Angles -

two angles whose sides are opposite rays

Complementary Angles -

two angles whose measures have a sum of 90 degrees

Supplementary Angles -

two angles whose measures have a sum of 180 degrees

Section 1.7

Midpoint and Distance in the Coordinate Plane

Midpoint Formula

On a Number Line

The coordinate of the midpoint is the average or mean of the coordinates of the endpoints.

On the Coordinate Plane

The coordinates of the midpoint are the average of the x-coordinates and the average of the y-coordinates of the endpoints.

Section 1.8

Perimeter, Circumference, and Area

Perimeter -

the distance around a polygon

Found by adding the lengths of the sides of the polygon

Area -

the number of square units a polygon encloses

Found in different ways depending on the type of polygon

Rectangles

Circles

**GOAL!**

**Chapter 1**

**Tools of Geometry**

Line -

represented by a straight path that extends in two opposite directions without end and has no thickness. A line contains infinitely many points.

Lines are named by any two points on the line or by a single lowercase letter

Plane -

represented by a flat surface that extends without end and has no thickness. A plane contains infinitely many lines.

Planes are named by either a single capital letter or by three points in the plant that are not on the same line.

**.**

**A**

read as "point A"

A B C

m

Collinear -

points that lie on the same line

Coplanar -

points and lines that lie on the same plane

Space -

the set of all points in three dimensions

Segment -

the part of a line that consists of two endpoints and all the points in between them

Segments are named by their two endpoints

J K

Ray -

part of a line that consists of one endpoint and all the points of the line on one side of the endpoint.

Named by the endpoint first and then another point in the direction of the ray

M N

Opposite Rays -

two rays that share the same endpoint and form a line

Named by using the shared endpoint and then one point in opposite directions from the shared endpoint

E F G

Postulate/Axiom -

an accepted statement of fact.

Much of Geometry relies on postulates. Consider them the building blocks of the logical system in Geometry.

Postulate 1-1

Through any two points, there is exactly one line.

Postulate 1-2

If two distinct lines intersect, then they intersect in exactly one point.

Postulate 1-3

If two distinct planes intersect, then they intersect in exactly one line.

Postulate 1-4

Through any three non-collinear points, there is exactly one plane

Homework #1

Section 1.2 - Points, Lines, and Planes

Available on Math XL (Due by

end of class

tomorrow)

The ruler postulate essentially allows us to find the distance between any two points.

A B

The

distance

between point A and point B is the

absolute value

of the difference of their coordinates.

Postulate 1-6

(Segment Addition Postulate)

If three points A, B, and C are collinear and B is between A and C, then AB + BC = AC

A B C

When two numerical expressions have the same value, we say they are equal (=).

Similarly, if two segments have the same length, we say they are

congruent

( )

Segments can be shown to be congruent by using a equal number of marks through them

Midpoint -

a point that divides a segment into two congruent segments

It's the exact middle

Bisect -

to cut a segment into two equal halves

Segment Bisector -

a point, line, ray, or segment that cuts another segment into two equal halves

Homework #2

Section 1.3 - Measuring Segments

Available on Math XL (Due by

end of class

tomorrow)

Types of Angles

Acute Angle -

less than 90 degrees

Right Angle -

equal to 90 degrees

Obtuse Angle -

greater than 90 degrees

Straight Angle -

equal to 180 degrees

Congruent Angles -

angles with the same measure.

You can show that angles are congruent by marking them with the same number of arcs.

Postulate 1-8 (Arc Addition Postulate)

If point B is in the interior of angle AOC, then:

m<AOB + m<BOC = m<AOC

A

O

B

C

Homework #3

Section 1.4 - Measuring Angles

Available on Math XL (Due by

end of class

tomorrow)

Finding Information from a Diagram

When a diagram has no marks or measures, there are only a few things you can conclude:

1. Angles are adjacent

2. Angles are supplementary

3. Angles are vertical Angles

You are

NOT

able to conclude:

1. Angles or segments are congruent

2. An angle is a right angle

3. Angles are complementary

Postulate 1-9 (Linear Pair Postulate)

If two angles form a linear pair, then they are supplementary.

A

B

C

D

Angle Bisector -

a ray that divides and angle into two congruent angles

A

O

B

C

Homework #4

Section 1.5 - Exploring Angle Pairs

Available on Math XL (Due by

end of class

tomorrow)

Midpoint Formula

Distance Formula

Allows you to find the distance between two points on the coordinate plane

Homework #5

Section 1.7 - Midpoint/Distance in Coordinate Plane

Available on Math XL (Due by

end of class

tomorrow)

Squares

s

s

s

s

P = 4s

A = s s

.

h

b

P = 2h + 2b

A = b h

.

r

d

r = radius

d = diameter

C = d -or- C = 2 r

A =

Triangles

a

b

c

P = a + b + c

h

Postulate 1-10 (Area Addition Postulate)

The area of a region is the sum of the areas of its non-overlapping parts.

In other words, if you have a combination of various shapes, the total area is the area of each individual shape added together.

Homework #6

Section 1.8 - Perimeter, Circumference, and Area

Available on Math XL (Due by

end of class

tomorrow)

Notebook Question #1

Name the line in 3 ways -->

Name 5 different segments -->

Name 4 different rays -->

Name a pair of opposite rays -->

Notebook Question #2

Name 3 Planes

Where do TS and SW intersect?

Describe the intersection of the front and the bottom of the prism.

Notebook Question #3

Name 3 collinear points

Where do the two lines intersect?

Are points C, B, and G collinear?

Notebook Question #1

What is the length of AC?

What is the length of BE?

Name the two congruent segments.

Notebook Question #2

RS = 8y + 4

ST = 4y + 8

Solve for y

RT = 15y - 9

Notebook Question #3

A is the midpoint of XY.

Solve for x.

Notebook Question #1

Name the angle in 4 distinct ways.

Notebook Question #2

Name an acute angle. Name a right angle.

Name an obtuse angle. What is the measure of <CAD?

Notebook Question #3

<TQR is a straight angle. Solve for x.

Notebook Question #1

Name an angle that is

complementary

to <EOD.

Name an angle that is

supplementary

to <AOE.

Name an angle that is

vertical

to <COD.

Name 2 angles that are

adjacent

to <DOE.

Notebook Question #2

GH bisects <FGI. Solve for x.

Notebook Question #3

<RQS and <TQS are a

linear pair

.

<RQS = 2x + 4

<TQS = 6x +20

Solve for x and find the measure of each angle.

Notebook Question #1

Find the

distance

between each pair of points:

J(2,-1) and K(2,5) R(0,5) and S(12,3)

Notebook Question #2

Find the coordinates of the

midpoint

of AB

A(7,10) B(5,-8) A(13,8) B(-6,-6)

Notebook Question #3

The coordinates of point T are given. The midpoint of ST is (5,-8).

Find the coordinates of point S

.

T (1,12)

Midpoint (5,-8)

Notebook Question #1

Find the perimeter and area of the rectangle.

Notebook Question #2

Find the circumference and area of the circle.

Notebook Question #3

Find the total perimeter and area of the figure.