**Learning Objectives**

After completing this tutorial, you should be able to:

- Know what a line is.
- Identify the different types of angles.
- Find a complimentary angle to a given angle.
- Find a supplementary angle to a given angle.
- Know what a polygon is.
- Identify types of triangles.
- Know what a quadrilateral is.
- Find the value of a corresponding angle or side given congruent figures.
- Find the value of a corresponding angle or side given similar figures.
- Know the relationship of the angles formed when a transversal cuts through two parallel lines.
- Know the parts of a circle.
- Tell the difference between inscribed and circumscribed.

** Introduction**

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**

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).

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.

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:**

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:**

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

For example, if you have a triangle, which has 3 sides,
the sum of the
measures of the interior angles would be

**The area of a polygon is the measure of the enclosed
interior.**

First of all, we need to know how many sides we are
dealing with.
How many sides are there on a pentagon? If you said 5, you are
right
on!!

Putting 5 into the **sum
of the measures
of the interior angles of an n-sided
polygon
formula we get:**

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 viola .

The **hypotenuse** is the side opposite the right angle and the **legs** are the sides that meet at the right angle.

Some of the more common quadrilaterals are

Note that a **square** is a special type of
rectangle, one in which
all four sides are equal to each other

Note that **rectangles** are a
special type of
parallelograms.

**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.

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.**

If C = 50 degrees and E = 75 degrees, what is the measure of J?

If AB = 20, BC = 30, and CD = 40, then what is the length of GF?

Note: Corresponding angles and sides are found by
matching up the letters
of each figure’s name in the order that they are listed. A
corresponds
with F, B corresponds with G and so forth.

Since J corresponds
with E
and the
figures are congruent, **then J
= E = 75
degrees.**

Since side GF (or FG) corresponds to side AB and the
figures are congruent, **then
side GF (or FG) = side AB = 20.**

**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.

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.**

If C = 25, E = 40, and G = 30, what is the measure of L?

If AG = 5, HN = 20, and BC = 40, what is the length of IJ?

Since L corresponds
to E and
the
figures are similar,** then L
= E = 40.**

Since side IJ corresponds to side BC and the figures are similar to each other, then IJ and BC are in proportion to each other. Similarly, AG and HN are in proportion to each other.

When setting up the proportion, make sure that you set it up the same on both sides.

***Cross multiply**

**alternate interior angles are
equal.**

Alternate interior angles are interior angles on
opposite sides of
the transversal.

4 and 5 of the above diagram are alternate interior angles.

3 and 6 of the above diagram are also alternate interior angles.

**alternate exterior angles are
equal. **

Alternate exterior angles are are exterior angles
opposite sides of
the transversal.

1 and 8 of the above diagram are alternate exterior angles.

2 and 7 of the above diagram are also alternate exterior angles.

**corresponding angles are equal. **

Corresponding angles are one interior and one
exterior angle that are
on the same side of the transversal.

1 and 5 of the above diagram are corresponding angles.

2 and 6 of the above diagram are also corresponding angles.

3 and 7 of the above diagram are also corresponding angles.

4 and 8 of the above diagram are also corresponding angles.

Find 8 if 1 = 120 degrees.

Find 4 if 5 = 120 degrees.

Find 7 if 3 = 60 degrees.

Find 3 if 5 = 120 degrees.

Since 8 and 1
are alternate exterior angles and the two lines are parallel, **then 8
= 1 = 120
degrees.**

Since 4 and 5
are alternate interior angles and the two lines are parallel, **then 4
= 5 = 120
degrees.**

Since 7 and 3
are corresponding angles and the two lines are parallel, **then 7
= 3 = 60
degrees.**

Since 3 and 5 are not alternate exterior, alternate interior or corresponding angles, they are not guaranteed to be equal.

However, since 3 and 4 make a straight angle (180 degrees) and 4 and 5 are alternate interior angles (which means they are equal), we can find the measure of 3.

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.

The tangent line and the radius of the circle that has an endpoint at the point of tangency are perpendicular to each other.

In this situation we can also say that the circle is circumscribed about the polygon.

In this same situation we can say that the circle is inscribed in the polygon.

** Practice Problems**

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 complementary angles.

1a. What is the complementary angle to 33 degrees?

(answer/discussion to 1a)

(answer/discussion to 1a)

Practice Problem 2a:Answer the question on supplementary angles.

2a. What is the supplementary angle to 33 degrees?

(answer/discussion
to 2a)

Practice Problem 3a:Answer the question on congruent figures.

3a. Figure ABCD is congruent to figure EFGH

If B = 55, C = 45, and D = 30, what is the measure of G?

If AD = 10, EF = 15, and BC = 12, what is the length of
EH?

(answer/discussion
to 3a)

Practice Problem 4a:Answer the question on similar figures.

4a. Figure ABCDE is similar to figure FGHIJ.

If A = 30, C = 40, and E = 50, what is the measure of F?

If AE = 10, FJ = 20, and BC = 40, what is the length of
GH?

(answer/discussion
to 4a)

Practice Problems 5a - 5d:Use the following figure to answer the questions

5a. Find 2
if 7 = 50
degrees.

(answer/discussion to 5a)

(answer/discussion to 5a)

5b. Find 3
if 6 = 50
degrees.

(answer/discussion to 5b)

(answer/discussion to 5b)

5c. Find 1
if 5 =
130 degrees.

(answer/discussion to 5c)

(answer/discussion to 5c)

5d. Find 4
if 6 = 50
degrees.

(answer/discussion to 5d)

(answer/discussion to 5d)

** Need Extra Help on these Topics?**

**http://www.mathleague.com/help/geometry/polygons.htm**

This webpage goes over the polygons and circles.

**Go to Get
Help Outside the
Classroom found in Tutorial 1: How to Succeed in a Math Class for
some
more suggestions.**

Last revised on August 6, 2011 by Kim Seward.

All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.