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G E O M E T R Y
CHAPTER 1
BASICS OF GEOMETRY
Notes & Study Guide
2
TABLE OF CONTENTS
PATTERNS AND INDUCTIVE REASONING ...................................................... 3
POINTS, LINES AND PLANES ........................................................................... 6
SEGMENTS AND THEIR MEASURES ............................................................... 9
ANGLES AND THEIR MEASURES................................................................... 11
SEGMENT AND ANGLE BISECTORS ............................................................. 13
ANGLE PAIR RELATIONSHIPS ....................................................................... ##
INTRODUCTION TO AREA & PERIMETER ..................................................... ##
HOMEWORK EXAMPLES ................................................................................ ##
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PATTERNS & INDUCTIVE REASONING
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Geometry was originally developed when people began to recognize and
describe patterns they found in the shapes around them.
As in other math, some of these patterns are numerical; but most of them are
visual.
Both types are based on numbers, but are represented differently.
In this class, your job is to be able to recognize when a pattern appears
and to be able to either describe it or use it to solve a problem.
DESCRIBING NUMERICAL PATTERNS
Numerical patterns used in Geometry will be similar to those in Algebra.
To describe one: look for a value that is being added, subtracted, multiplied or
divided from one value to the next.
EXAMPLES
1, 4, 16, 64, ___, ____
description:
0, 1, 4, 9, ___, ___
description:
-5, -2, 4, 13, ___, ___
description:
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PATTERNS AND INDUCTIVE REASONING
DESCRIBING VISUAL PATTERNS
Visual patterns are a little different, but not difficult.
You are still looking for some kind of numerical pattern, but it will be represented
by physical objects or images (not actual numbers).
Examine the figures to find the pattern and then apply it to the sequence.
You’ll need to draw it out to continue the pattern. Include any colors or shading!
EXAMPLES
USING INDUCTIVE REASONING
In Geometry, we often look for a pattern and use it to make a decision or draw a
conclusion. This is called inductive reasoning.
The decision/conclusion that you make is called a conjecture.
Conjecture: an unproven statement that is based on observations
STEPS FOR INDUCTIVE REASONING
1. Look for a pattern (numeric or visual)
2. Make a conjecture (use the pattern to draw a conclusion)
3. Verify the conjecture (confirm that it is true; chapter 2 and beyond)
EXAMPLE: Draw the next image in this sequence.
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PATTERNS AND INDUCTIVE REASONING
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USING INDUCTIVE REASONING (cont.)
Remember, a conjecture is unproven; like a hypothesis in science. You still have
to prove if you’re right or wrong. It can be tough and there are many methods.
For Geometry, the guidelines are..
(1) To prove a conjecture true, you must prove that it’s true for ALL cases!
(2) To prove a conjecture false, you need to provide a single counterexample.
counterexample: a single example that proves a statement is false
EXAMPLES: Show each statement is false by providing a counterexample.
 “All prime numbers are odd.”
 “The square root of a number is always less than the number itself.”
 “All sophomores are right-handed.”
It is always easier to prove a statement is false (one time) than it is to prove that
it’s true (every time).
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POINTS, LINES AND PLANES
THE BUILDING BLOCKS OF GEOMETRY
“Building blocks” refers to the core set of individual pieces of Geometry. These
pieces are what all other shapes, figures and solids are made from.
You will need to be able to sketch, name and identify these building blocks using
the standard notation techniques.
POINTS
A point is an individual location in space that has no
dimension or size. Points are represented as dots when
drawn.
Different points are identified by their labels, which are
capital letters of the alphabet written net to the dot.
NAMING: points are named by their labels (“point A”, “point B”, etc.)
SYMBOL NOTATION: use the naming above or the label letter (nothing else)
SEGMENTS
A segment is a collection of individual points that are arranged
and connected in a straight path.
Segments both begin and end at specific locations called
endpoints.
NAMING: segments are named by their endpoints; order does not matter
(“segment AB or BA”, “segment GH or HG”, etc.)
SYMBOL NOTATION: use the endpoints with a small segment
drawn above
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AB BA
POINTS, LINES AND PLANES
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LINES
A line is a collection of individual points that are arranged
and connected in a never-ending straight path.
Lines have no starting/stopping points (no endpoints). They are drawn with
arrows on each end to show this.
NAMING: lines are named by identifying 2 points on it or by a lowercase letter
(“line AB or BA”, “line XY or YX”, etc.)
NOTATION: use the lowercase letter with the word “line”. (“line g”, “line c”, etc.)
SYMBOL NOTATION: use the 2 points with a small line drawn
above
AB BA
RAYS
A ray is a collection of individual points that are arranged
and connected in a never-ending straight path with a
specific starting point.
Rays have a starting point (1 endpoint) but do not stop. They are drawn with a
dot on one end and an arrow on the other.
NAMING: rays are named using the endpoint & a point on it (IN THAT ORDER!)
(“ray AB”, “ray BA”, etc.)
SYMBOL NOTATION: use the 2 points (endpoint first!) with
a small ray drawn above.
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BA
AB
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POINTS, LINES AND PLANES
PLANES
A plane is collection of individual points that are
arranged and connected to create a never-ending flat
surface.
Planes are FLAT! They extend in 2 dimensions (planar). WE draw them to look
like a slanted rectangle. The slant shows the dimension.
NAMING: planes are named using any 3 identified points that lie on the plane.
(“plane ABC”, “plane XYZ”, etc.)
NOTATION: use the 3 points and the word “plane” in front (no symbol used).
OTHER IMPORTANT TERMS
Collinear points – any set of points (2 or more) that
lie on the same line
Coplanar points – any set of points (2 or more) that lie
on the same plane (surface)
Opposite rays – two rays with the same endpoint
that form a line
Intersection – two or more figures intersect if they have one or more points in
common. The intersection is only the point(s) that are being shared.
The most common are…
2 or more lines/segments
a line/segment & a plane
2 or more planes
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SEGMENTS AND THEIR MEASURES
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This section (1.3) is the first to use postulates. They provide the framework for
all that we will do in this class. There are 29 that you must learn (page 827).
postulate – a rule that is accepted to be true without proof
THE RULER POSTULATE
The endpoints of a segment can be
matched to any two values on a
number line or a ruler.
The values that the endpoints line up with are called the coordinates.
The Ruler Postulate states that the distance between the two endpoints is the
same as the absolute value of the difference of the two coordinates.
In other words, if you simply take the larger coordinate and subtract the smaller,
you will get the distance.
This distance, written as AB, is also called the length of the segment.
EXAMPLE:
Use a ruler to measure the segment to the nearest
millimeter. CD = ______
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SEGMENTS & THEIR MEASURES
SEGMENT ADDITION POSTULATE
When three collinear points lie on a line, there
will always be one point between the other two
(shown).
When this happens, the long segment (AC) is
split into two smaller segments (AB and BC).
The Segment Addition Postulate says that the sum of the two smaller segment
lengths is the same as the overall larger segment. (AB + BC = AC)
This Postulate also applies to 3 or more
segments as long as the points are
collinear (shown).
THE DISTANCE FORMULA
Sometimes segments will exist in a coordinate plane
(shown). The Distance Formula is used to calculate
the distance between the endpoints.
In the formula the variables (x1, x2, y1, y2) refer to the x
and y coordinates of each endpoint.
EXAMPLE: What is the length of AC? _______
Segments with the same length are called congruent.
The symbol is an equal sign with a “squiggle” on top.
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ANGLES & THEIR MEASURES
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ANGLES
An angle is formed when two rays share the same
endpoint (initial point).
The rays form the sides of the angle; while the
shared endpoint is called the vertex.
NAMING: angles are named using 3 points; 1 from each ray and the vertex
(“angle ABC”, “angle XYZ”, “angle RST”, etc.)
IMPORTANT! The vertex point MUST be in the middle of the name!
SYMBOL NOTATION: use the naming procedure above
with a small angle drawn in front
Angles can also be named by vertex only if it’s the only one using that
vertex.
THE MEASURE OF AN ANGLE
The measure on an angle refers to how far apart the two rays (sides) are. The
closer the rays are to each other, the smaller the measure is.
Angles are measured in degrees and the tool of choice is the protractor.
To measure an angle…
(1) Line up the vertex and one ray to the zero line
(2) Read the value where the other sides intersects the protractor
When two angles have the same
measure, then they are said to be
congruent.
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ANGLES & THEIR MEASURES
OTHER IMPORTANT TERMS
Interior – the region of an angle located in
between the two sides
Exterior – the region of an angle located
outside the two sides
ANGLE ADDITION POSTULATE
If an angle has an extra ray within its’ interior, then
that angle is split into two smaller angles (shown).
The Angle Addition Postulate states that the total
of the measures of the two smaller angles is the
same as the measure of the larger angle.
Notice how the two smaller angles are sharing side SP and vertex S.
Angles that share a common vertex and side are called adjacent angles.
EXAMPLES: Find the measure of the unknown angle.
CLASSIFYING ANGLES
Angles are classified according to their measures. There are 4 categories…
Acute – angle with a measure between 0° and 90°
Right – angle with a measure of exactly 90°
Obtuse – angle with a measure between 90° and 180°
Straight – angles with a measure of exactly 180°
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SEGMENT AND ANGLE BISECTORS
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The word bisect means “to cut into two equal pieces. Therefore, any line,
segment or ray that will cut a segment or angle into two equal pieces is referred
to as a bisector.
SEGMENT BISECTORS
A segment bisector is any line, segment
or ray that intersects a segment AND splits
it into two equal pieces.
The point where the bisection occurs is
called the midpoint (literally means “the
point in the middle).
NOTATION: It might not be easy to see when a segment is bisected. To help, we
use tick marks. Small matching lines that show two or more segments are
congruent (the little red marks shown).
THE MIDPOINT FORMULA
Sometimes segments will exist on a coordinate
plane (shown). The Midpoint Formula is used
to calculate the coordinates of its’ midpoint.
To use the Midpoint Formula…
(1) add the x-values of the endpoints and divide it by 2.
(2) add the y-values of the endpoints and divide it by 2.
The midpoint is the ONLY spot where a segment bisector touches a segment.
EXAMPLES
C(10, 8) and D(–2, 5)…what is the midpoint, M?
E(4, 4) and F(4, –18)…what is the midpoint, M?
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SEGMENT AND ANGLE BISECTORS
ANGLE BISECTORS
An angle bisector is any line, segment or ray that
intersects an angles’ vertex AND splits that angle into
two congruent adjacent angles.
CD is the angle bisector of angle ACB (shown)
NOTATION: To show an angle has been bisected, congruence marks (small
curves) are placed in the angles’ interior near the vertex.
Congruence marks are the angle version of tick marks.
When a angle is bisected, the sum of the two smaller angle measures is equal to
the measure of the whole angle. (Variation of the Angle Addition Postulate)
EXAMPLES:
QS is an angle biesctor in each, find the measures of the other two angles.
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ANGLE PAIR RELATIONSHIPS
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In the last section, we worked with individual angles and their measures. In this
section, we will look at classifying angles when they appear in pairs.
VERTICAL ANGLES
Vertical angles occur when the sides of two angles
form two pairs of opposite rays (or 2 lines).
In other words, vertical angles appear when two
lines make an “X”.
The vertical angles are the ones that are directly across from each other, so they
will always come in pairs.
Vertical angles are ALWAYS congruent!
LINEAR PAIR
Linear pairs occur when two angles are adjacent
(next to) each other and the unshared sides form a
line.
In other words, it’s a straight line with a ray
sticking out of it.
The linear pair must consist of 2 angles (duh!) and they must be connected to
each other.
The angles in a linear pair will ALWAYS total 180°!
COMPLEMENTARY & SUPLEMENTARY ANGLES
Two angles are complementary if the sum of their measures is exactly 90°.
Two angles are supplementary if the sum of their measures is exactly 180°.
These types of angles DO NOT have to be connected to each other. As long as
they total the right amount, that’s all that counts.
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SEGMENT AND ANGLE BISECTORS
EXAMPLES:
If angle 8 = 65°, then angle 6 = ______
If angle 9 = 104°, then angle 8 = ______
Find the value of the variable in each figure using linear pairs or vertical angles.
The two angles listed are complementary. Find the measure of each.
The two angles listed are supplmentary. Find the measure of each.
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INTRO TO PERIMETER, CIRCUMFERENCE & AREA
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In Geometry, we are often interested in finding out details about the
shapes/figures we interact with. Area, perimeter and circumference are some of
the common ones.
PERIMETER
Perimeter is the total distance around the outside of any planar (flat) figure. It is
calculated by simply adding all of the figures’ sides together.
CIRCUMFERENCE
Circumference is the total distance around the outside of a
circle. Since circles, have no sides, circumference is
calculated using a formula.
C = 2p r
where r = radius &
= 3.14
AREA
Area is the amount of space that a planar (flat) figure occupies. It is calculated
using a unique formula for each figure. Area is measured in square units!
Square
A = s2
Rectangle
A=L•W
Triangle
A = (1/2)bh
Circle
A = r2
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INTRO TO PERIMETER, CIRCUMFERENCE & AREA
EXAMPLES:
Find the perimeter (or circumference) AND the area of each figure. Use 3.14 for
when necessary. Round any answers to the nearest tenth.
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ADDITIONAL EXAMPLES
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ADDITIONAL EXAMPLES