Learning Objectives
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
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.
Let’s set it up and solve it algebraically, letting x be the missing angle and see what we get:
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.
Putting 5 into the sum of the measures of the interior angles of an n-sided polygon formula we get:
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 .
Types of TrianglesSome 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
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.
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?
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 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.
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.
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.
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.
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 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
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.
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.
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
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.
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