Beginning Algebra Tutorial 33


Beginning Algebra
Tutorial 33: Basic Geometry


WTAMU > Virtual Math Lab > Beginning Algebra

 

deskLearning Objectives


 
After completing this tutorial, you should be able to:
  1. Know what a line is.
  2. Identify the different types of angles.
  3. Find a complimentary angle to a given angle.
  4. Find a supplementary angle to a given angle.
  5. Know what a polygon is.
  6. Identify types of triangles.
  7. Know what a quadrilateral is.
  8. Find the value of a corresponding angle or side given congruent figures.
  9. Find the value of a corresponding angle or side given similar figures.
  10. Know the relationship of the angles formed when a transversal cuts through two parallel lines.
  11. Know the parts of a circle.
  12. Tell the difference between inscribed and circumscribed.




desk 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.

 

 

desk Tutorial


 
 

Lines
 
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:

line


 
The symbol line ab, which includes the arrow heads at both ends indicates the whole line where line ab, 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).

 
  Angles
 
When two lines intersect at one point, they form four angles as shown below:

vertical angles


 
The opposite angles of the figure above are called vertical angles. 

Vertical angles have the same measure. 

From the illustration above, angle apcand angle dpbare vertical angles and would have the same measurement.

angle apd and angle cpbare another set of vertical angles on this illustration and would have equal measurements.


 
  Types of Angles
 
A right angle is one that measures exactly 90 degrees:

right angle


 
An acute angles is one that measures between 0 degrees and 90 degrees:

acute angle


 
An obtuse angle is one that measures between 90 degrees and 180 degrees:

obtuse angle


 
A straight angle is one that measures exactly 180 degrees:

straight angle


 
  Complementary Angles
 
Complementary angles are two angles whose sum measures 90 degrees.

 
 
notebook 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:


 
example 1
*Complimentary angles sum up to be 90

 


 
The complimentary angle to 68 degrees is 22 degrees.

 
  Supplementary Angles
 
Supplementary angles are two angles whose sum measures 180 degrees.

 
 
notebook 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:


 
example 2
*Supplementary angles sum up to be 180

 
The supplementary angle to 125 degrees is 55 degrees.

 
  Parallel Lines 
 
Parallel lines have the same slope:

parallel


 
  Perpendicular Lines 
 
Perpendicular lines intersect at right angles:

perpendicular


 

Polygons
 
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


  Triangle (3 sides)
triangle Quadrilateral (4 sides)
quadilateral
 
The sum of the measures of the interior angles of an n-sided polygon is polygon.

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


 
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.

 
 
notebook 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?

 
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:

polygon


 
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 polygon.


 

Types of Triangles
 
  The following three types of triangles are categorized by their angles:
 
An acute triangle is a triangle that has three acute angles:

acute triangle


 
An obtuse triangle is a triangle that has one obtuse angle:

obtuse triangle


 
A right triangle is a triangle that has a right angle:

rigth triangle

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:

isoceles triangle


 
An equilateral triangle is one that has three equal sides:

equilateral triangle


 
A scalene triangle is a triangle where no two sides are equal in length:

scalene triangle


 

Quadrilaterals
 
A quadrilateral is a four sided polygon.

Some of the more common quadrilaterals are 


  Rectangle
 

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:

rectangle

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


  Parallelogram
 

A parallelogram is a quadrilateral in which opposite sides are parallel and have equal length and opposite interior angles have the same measure: 

parallelogram

Note that rectangles are a special type of parallelograms. 


 
  Trapezoid
 
A trapezoid is a quadrilateral in which one pair of opposite sides are parallel:

trapezoid


 
  Congruent Figures
 
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:

congruent

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.


 
 
notebook Example 4:   Figure ABCDE is congruent to figure FGHIJ. 

If angleC = 50 degrees and angleE = 75 degrees, what is the measure of angleJ? 

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 angleJ corresponds with angleE and the figures are congruent, then angleJ = angleE = 75 degrees.
 

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


 
 
  Similar Figures
 
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:

similiar triangles

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.


 
 
 
notebook Example 5:   Figure ABCDEFG is similar to figure HIJKLMN. 

If angleC = 25, angleE = 40, and angleG = 30, what is the measure of angleL? 

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


 
Since angleL corresponds to angleE and the figures are similar, then angleL = angleE = 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.


 
example 4
*Corresponding sides of similar figures are in proportion to each other
 
 
 

*Cross multiply
 
 
 

 


 
Side IJ = 160.

 
  Parallel Lines 
Cut by a Transversal
  parallel
 
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.
 

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

angle4 and angle5 of the above diagram are alternate interior angles.

angle3 and angle6 of the above diagram are also alternate interior angles.


 
If two parallel lines in a plane are intersected by a transversal, then

alternate exterior angles are equal.
 

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

angle1 and angle8 of the above diagram are alternate exterior angles.

angle2 and angle7 of the above diagram are also alternate exterior angles.


 
If two parallel lines in a plane are intersected by a transversal, then

corresponding angles are equal.
 
 

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

angle1 and angle5 of the above diagram are corresponding angles.

angle2 and angle6 of the above diagram are also corresponding angles.

angle3 and angle7 of the above diagram are also corresponding angles.

angle4 and angle8 of the above diagram are also corresponding angles.


 
notebook Example 6:   Given the following parallel lines cut by a transversal
parallel

Find angle8 if angle1 = 120 degrees.

Find angle4 if angle5 = 120 degrees. 

Find angle7 if angle3 = 60 degrees.

Find angle3 if angle5 = 120 degrees.


 
Find angle8 if angle1 = 120 degrees.

Since angle8 and angle1 are alternate exterior angles and the two lines are parallel, then angle8  = angle1 = 120 degrees.


 
Find angle4 if angle5 = 120 degrees.

Since angle4 and angle5 are alternate interior angles and the two lines are parallel, then angle4 = angle5 = 120 degrees.


 
Find angle7 if angle3 = 60 degrees.

Since angle7 and angle3 are corresponding angles and the two lines are parallel, then angle7 = angle3 = 60 degrees.


 
Find angle3 if angle5 = 120 degrees.

Since angle3 and angle5 are not alternate exterior, alternate interior or corresponding angles, they are not guaranteed to be equal. 

However, since angle3 and angle4 make a straight angle (180 degrees) and angle4 and angle5 are alternate interior angles (which means they are equal), we can find the measure of angle3.


 
example 5
*Alternate interior angles are =
*Straight angle = 180
 

 


 
angle3 = 60 degrees.


 

Circles
 
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.

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. 

circle
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.


 
  Tangent to a Circle
 
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.

tangent


 
 
  Circumscribed and Inscribed
 
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.

inscribed


 
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.

inscribed


 
  Concentric Circles
 
Concentric circles are two or more circles that share the same center.

concentric


 

 
desk 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.

 

pencil Practice Problem 1a: Answer the question on complementary angles.

 

1a.  What is the complementary angle to 33 degrees?
(answer/discussion to 1a)


 

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

 

2a.  What is the supplementary angle to 33 degrees?
(answer/discussion to 2a)

 

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

 

3a.  Figure ABCD is congruent to figure EFGH 

If angleB = 55, angleC = 45, and angleD = 30, what is the measure of angleG? 

If AD = 10, EF = 15, and BC = 12, what is the length of EH?
(answer/discussion to 3a)

 

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

 

4a.  Figure ABCDE is similar to figure FGHIJ.

If angleA = 30, angleC = 40, and angleE = 50, what is the measure of angleF? 

If AE = 10, FJ = 20, and BC = 40, what is the length of GH?
(answer/discussion to 4a)

 

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

problem 5

 

5a.   Find angle2 if angle7 = 50 degrees.
(answer/discussion to 5a)
5b.  Find angle3 if angle6 = 50 degrees.
(answer/discussion to 5b)


5c.  Find angle1 if angle5 = 130 degrees.
(answer/discussion to 5c)
5d.  Find angle4 if angle6 = 50 degrees.
(answer/discussion to 5d)

 

 

 

 

desk Need Extra Help on these Topics?



 
The following is a webpage that can assist you in the topics that were covered on this page:
 

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.


 


WTAMU > Virtual Math Lab > Beginning Algebra


Last revised on August 6, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.