Beginning Algebra
Tutorial 33:
Basic Geometry
 Learning Objectives
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After completing this tutorial, you should be able to:
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Know what a line is.
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Identify the different types of angles.
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Find a complimentary angle to a given angle.
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Find a supplementary angle to a given angle.
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Know what a polygon is.
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Identify types of triangles.
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Know what a quadrilateral is.
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Find the value of a corresponding angle or side given congruent figures.
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Find the value of a corresponding angle or side given similar figures.
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Know the relationship of the angles formed when a transversal cuts
through
two parallel lines.
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Know the parts of a circle.
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Tell the difference between inscribed and circumscribed.
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Introduction
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| In this tutorial we will be looking at basic concepts
of geometry.
This lesson is designed to get you familiar with the terminology used
in
some basic geometry problems. We will be looking at lines, angles,
polygons,
triangles, quadrilaterals, congruent figures, similar figures, parallel
lines, and circles. I guess you better get to it. |
Tutorial
|
| A line extends infinitely and is named by labeling
two points on
the line with capital letters or by putting a lower case letter near
it.
Both are illustrated below:
|
The symbol  ,
which includes the arrow heads at both ends indicates the whole line
where  ,
which does not have the arrow heads, indicates a line segment, which is
finite in length (only the part of the line from A to B). |
| When two lines intersect at one point, they form
four angles as
shown below:
|
| The opposite angles of the figure above are called
vertical angles.
Vertical angles have the same measure.
From the illustration above,  and  are
vertical angles and would have the same measurement.
and  are
another set of vertical angles on this illustration and would have
equal
measurements.
|
| A right angle is one that
measures exactly
90 degrees:
|
| An acute angles is one that
measures between
0 degrees and 90 degrees:
|
| An obtuse angle is one that
measures between
90 degrees and 180 degrees:
|
| A straight angle is one that
measures exactly
180 degrees:
|
| Complementary angles are two angles whose sum
measures 90 degrees. |
 Example
1: What is the complementary angle to 68
degrees? |
| Basically we need an angle that when adding it to 68 we
get 90.
Let’s set it up and solve it algebraically,
letting x
be the missing angle and see what we get:
|
 |
*Complimentary angles sum up
to be 90
|
| The complimentary angle to 68 degrees is 22 degrees. |
| Supplementary angles are two angles whose sum
measures 180 degrees. |
| Example
2: What is the supplementary angle to 125
degrees? |
| Basically we need an angle that when adding it to 125
we get 180.
Let’s set it up and solve it algebraically, letting x
be the missing angle and see what we get:
|
 |
*Supplementary angles sum up
to be 180 |
| The supplementary angle to 125 degrees is 55 degrees. |
| Parallel lines have the same slope:
|
| Perpendicular lines intersect at right angles:
|
| A polygon is a closed figure composed of three or
more line segments
that intersect at their endpoints.
Each line segment is called the side.
The intersection at the endpoints is called the vertex.
Keep in mind that the number of sides and number of
interior angles
of a polygon are the same.
Polygons are named by the number of sides they
have.
Two common polygons are
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Triangle (3 sides)
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Quadrilateral (4 sides)
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The sum of the measures of the interior angles of an
n-sided
polygon is  .
For example, if you have a triangle, which has 3 sides,
the sum of the
measures of the interior angles would be
|
| The perimeter of any polygon is simply the sum of
all the sides
of that polygon.
The area of a polygon is the measure of the enclosed
interior.
|
| A regular polygon is one in which all of the
interior angles have
the same measure and all of the sides have the same length. |
| Example
3: What would be the sum of the measures of the
interior angles of a pentagon? What would be the measure of each
interior angle of a regular pentagon? |
| So for any pentagon, whether it is regular or not, the
sum of the measures
of the interior angles is 540 degrees.
Next we need to figure out what would be the measure of
each interior
angle of a regular pentagon.
Since we are talking specifically about a regular
pentagon, that means
all interior angles have the same measure. And since the total of
those measures is 540, what do you think the measure of each interior
angle
is? If you said 108 degrees give yourself a high five.
Just divide the total, 540, by the number of angles, in
this case 5
and voila  .
|
|
The following three types of
triangles are categorized
by their angles:
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| An acute triangle is a
triangle that has three
acute
angles:
|
| A right triangle is a triangle
that has a
right
angle:
The hypotenuse is the side opposite the right angle and the legs
are the sides that meet at the right angle. |
|
The following three types of
triangles are categorized
by their sides:
|
| An isosceles triangle is a
triangle that has
two equal sides:
|
| An equilateral triangle is one
that has three
equal sides:
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| A scalene triangle is a
triangle where no
two sides are equal in length:
|
| A quadrilateral is a four sided polygon.
Some of the more common quadrilaterals are
|
| A rectangle is a quadrilateral
in which the
opposite sides are equal in length and parallel to each other and the
four
interior angles are each 90 degrees:
Note that a square is a special type of
rectangle, one in which
all four sides are equal to each other
|
| A parallelogram is a quadrilateral in which opposite
sides are parallel
and have equal length and opposite interior angles have the same
measure:
Note that rectangles are a
special type of
parallelograms.
|
| A trapezoid is a quadrilateral in which one pair of
opposite sides
are parallel:
|
| Corresponding angles of congruent figures have the
same measure.
Corresponding sides of congruent figures are equal in
length.
Note: Corresponding angles and sides are found by
matching up the letters
of each figure’s name in the order that they are listed.
|
| Below is an example of two figures that are
congruent to each other:
Note how A corresponds with F, B corresponds with E, C
corresponds with
H and D corresponds with G.
In other words, side AB is the same as side FE, side BC
is the same
as side EH, side CD is the same as side HG and side DA is the same as
side
GF.
Matching up the corresponding points, the proper way
of to say this
is figure ABCD is congruent to figure FEHG.
|
| Corresponding angles of similar figures have the
same measure.
Corresponding sides of similar figures are in
proportion to each
other.
Note: Corresponding angles and sides are found by
matching up the letters
of each figure’s name in the order that they are listed.
|
| Below is an example of two figures that are similar
to each other:
Note how A corresponds with D, B corresponds with E, C
corresponds with
F.
All of the sides are in proportion to each other.
In this example,
side AB is twice as large as side DE, side BC is twice as large as side
EF, and side CA is twice as large as side FD.
Matching up the corresponding points, the proper way
of to say this
is figure ABC is similar to figure DEF.
|
 |
*Corresponding sides of
similar figures
are in proportion to each other
*Cross multiply
|
Parallel Lines
Cut by a Transversal
|
| If two parallel lines in a plane are intersected by
a transversal,
then alternate interior angles are equal, alternate exterior angles are
equal and corresponding angles are equal. |
| If two parallel lines in a plane are intersected by
a transversal,
then
alternate interior angles are
equal.
|
| If two parallel lines in a plane are intersected by
a transversal,
then
alternate exterior angles are
equal.
|
| If two parallel lines in a plane are intersected by
a transversal,
then
corresponding angles are equal.
|
 |
*Alternate interior angles are
=
*Straight angle = 180
|
 3
= 60 degrees. |
| A circle is a set of points that are equidistant
from a fixed point
called the center.
The radius (r on the diagram below) is
the distance from
the center of the circle to any point on the circle and can be shown as
a line segment connecting the center to a point on the circle.
The diameter is a line segment that connects two
points on the
circle and goes through the center of the circle. It is always
twice
as long as the radius.
A chord (line segment PQ on the diagram
below) is any
line segment whose endpoints are any two points on the circle.
The circumference of a circle is the distance
around the circle.
|
| An arc of a circle is the set of all points
between and including
two given points. One way to measure it is in degrees. Keep
in mind that the whole circle is 360 degrees.
When naming an arc, it is best to use three points - the
two endpoints
and a point in between - versus just the two endpoints. The
reason
is you can go clockwise or counterclockwise, which can make a
difference
when looking at the length of an arc.
Arc ADC would start at point A and go clockwise through D and end at
C.
Arc ADC is a 95 degree arc.
Arc ABC would start at point A and go counterclockwise
through B and
end at C. Since a circle is 360 degrees, then Arc ABC is a 360 -
95 = 265 degree arc.
|
| A line is tangent to a circle if it intersects the
circle at exactly
one point.
The tangent line and the radius of the circle that has
an endpoint at
the point of tangency are perpendicular to each other.
|
|
Circumscribed and Inscribed
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| A polygon is inscribed in a circle if each vertex of
the polygon
is a point on the circle.
In this situation we can also say that the circle is
circumscribed about
the polygon.
|
| A polygon is circumscribed about a circle if each
side of the polygon
is tangent to the circle.
In this same situation we can say that the circle is
inscribed in the
polygon.
|
| Concentric circles are two or more circles that
share the same center.
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Practice Problems
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| These are practice problems to help bring you to the
next level.
It will allow you to check and see if you have an understanding of
these
types of problems. Math works just like
anything
else, if you want to get good at it, then you need to practice
it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.
In fact there is no such thing as too much practice.
To get the most out of these, you should work the
problem out on
your own and then check your answer by clicking on the link for the
answer/discussion
for that problem. At the link you will find the answer
as well as any steps that went into finding that answer.
|
 Practice
Problem 1a:
Answer the question on complimentary
angles.
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Practice
Problem 2a:
Answer the question on supplementary
angles.
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Practice
Problem 3a:
Answer the question on congruent
figures.
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|
Practice
Problem 4a:
Answer the question on similar
figures.
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Practice
Problems 5a - 5d:
Use the following figure to answer
the questions
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Need Extra Help on These Topics?
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All contents copyright (C) 2001 - 2008, WTAMU and Kim Seward. All rights reserved.
Last revised on July 25, 2003 by Kim Seward. |
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