Beginning Algebra.basic Geometry2

Beginning Algebra

Tutorial 33: Basic Geometry

WTAMU > Virtual Math Lab > Beginning Algebra

Learning 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.

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

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:

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).

Angles

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.

Types of Angles

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

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

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

Parallel lines have the same slope:

Perpendicular Lines

Perpendicular lines intersect at right angles:

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) Quadrilateral (4 sides)

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?

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 .

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:

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

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:

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

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:

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:

Note that rectangles are a special type of parallelograms.

Trapezoid

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

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:

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.

Example 4: Figure ABCDE is congruent to figure FGHIJ.

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.

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:

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.

Example 5: Figure ABCDEFG is similar to figure HIJKLMN.

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.

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

*Cross multiply

Side IJ = 160.

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.

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.



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

Find 8 if 1 = 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.

*Alternate interior angles are = *Straight angle = 180

3 = 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.

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.

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.

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.

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

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

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)

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)

5b. Find 3 if 6 = 50 degrees. (answer/discussion to 5b)

5c. Find 1 if 5 = 130 degrees. (answer/discussion to 5c)

5d. Find 4 if 6 = 50 degrees. (answer/discussion to 5d)

Documents

Beginning Algebra.basic Geometry2