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# Chapter 1

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## Josh Truax

on 24 August 2017

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#### Transcript of Chapter 1

Section 1.2
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

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
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
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
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
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.
Full transcript