The Angle Business and Beginning Geometry

The Angle Business and Beginning Geometry

By

Mr. White, Foundations of Mathematics Teacher

California, USA

Students are not studying some of the additional information about angles, after learning how to draw and label angles (two or more common rays with a common vertex that is measured in a counterclockwise manner with a protractor, etc.) Now, we are examining angles formed by lines that do not initially have a common vertex.

Vertical Angles –

Vertical angles are created when two lines or line segments intersect, creating angles that have a common sides and a common vertex.

The angles above are a pair of vertical angles. The angles are formed by a pair of intersecting.

• The angles have unique properties. These properties are:
• vertical angles are congruent, and
• the linear pair of angles formed by the vertical angles is supplementary.

Two or more pair of vertical angles creates a diagram that is not well described in Geometry textbooks.

I have added an additional line to the drawing above. We have two oblique lines that are intersected by another oblique line. We have eight angles that are created. Depending on the measurement of the angles that are created by the intersections of the three lines, it is possible to create four pair of congruent angles.

Corresponding angles and measurement

Geometry vocabulary uses the term corresponding angles to describe the angles that appear in the same location where the oblique and horizontal lines intersect. These corresponding angles appear in many aspects of city or home or business planning: the location of a streetlight, bathrooms, fire hydrants…all of these things are usually located at corresponding locations on various street blocks or floors of a structure.

If the corresponding angles have an equal measurement, then a unique situation is created, where the corresponding angles are equal in measurement. Additionally, the same side angles are all supplementary to the adjacent angles, and the angles that exist “diagonally” from each other are equal in measurement, or are congruent.

These problems are presented in textbooks in a very simplistic and mechanical manner. Very few, if any, textbooks approach the subject as I have above, or will below.

In the diagram above, the angle measuring 5x+3 degrees corresponds to the angle measuring 100 degrees. The angles are equal in measurement. Therefore,

5x-3 = 97.

Solving for x,

5x = 100

x = 20 degrees

Let us use the same set of angles and lines, and change the location of the constant and the variable terms.

Now, the angle measuring 5x-3 degrees is diagonal (or “adjacent”, in geometry jargon) to the angle measuring. All of the corresponding angles are congruent. Therefore, Therefore,

5x-3 = 97.

Solving for x,

5x = 100

x = 20.

Using a final example, we use the same three lines, with the same congruent angles, and, once again, change the locations of the angles measured 5x-3, and 97.

In this instance, the measured angles are ‘alternate exterior’ angles. Therefore, if the corresponding angles are congruent, then the two angles that are numbered below are congruent. Therefore, as in the last example,

Therefore,

5x-3 = 97.

Solving for x,

5x = 100

x = 20.

In this example, the two measured angles are same side interior angles. Their relationship is supplementary. Therefore,

5x-3+98= 180

5x + 95 = 180

5x = 85

x = 17

Students can use what I call the Zorro ‘move’ to remember the measures of the angles in such a diagram (parallel lines intersected by a transverse line).

The path of the arrow highlights which angles are congruent. This happens because we have two sets of vertical angles linked to each other via the transverse line. As long as the corresponding angles are congruent, the ‘line of Zorro’ will link congruent angles. It never fails.

Measuring Angles Using Algebra

Measuring Angles using Algebra is a topic in modern day Geometry that has no relationship to t he original Geometry, nor to the Geometry taught in the mid- to- later- parts of the 20th Century. Its purpose is to give students an opportunity to measure angles in Geometry using the multi-step equation skills they learned in Algebra.

For example, if I am given an acute angle, ABC, and I indicate that the angle is composed of two adjacent angles, ABD and DBC, then the measure of ABC = the measure of ABD plus the measure of DBC.

I can assign an algebraic value to both of the angles whose sum equals the larger angle. I will assign 2x and 3x to ABD and DBC, respectively. Therefore, the measure of ABC equals 2x + 3x or 5x.

Another example is in order: if I am given an obtuse angle, ABC, and I indicate that the angle is composed of two angles, ABE and EBC, then the measure of ABC = the measure of ABE plus the measure of EBC. I can assign the algebraic expression of (2x + 5) to angle ABE, and the algebraic expression of (x - 4) to angle EBC. Therefore, the value of angle ABC is (2x + 5) + (x - 4). This simplifies as (3x + 1).

The second way that Algebra is used to measure angles in Geometry is to assign a measurement value to the given angles, and the solving for the variable value assigned to individual angles.

Using the examples from the preceding paragraphs, this is relatively simple to describe. For example, if I am given an acute angle, ABC, and I indicate that the angle is composed of two adjacent angles, ABD and DBC, then the measure of ABC = the measure of ABD plus the measure of DBC.

I can assign an algebraic value to both of the angles whose sum equals the larger angle. I will again assign 2x and 3x to the measure of two adjacent angles, ABD and DBC, respectively. Therefore, the sum of the two angles equals 2x + 3x or 5x, which is the measure of the resultant angle, ABC. (I am using the term 'resultant angle' to describe the larger angle that is formed by the summing of the values of the two adjacent angle.) If I assign a degree value to ABC, namely 60, then the value of 2x and 3x is easily obtained. 2x + 3x = 5x. 5x = 60. x = 12. Therefore, the measure of angle ABD equals 24 degrees, and the measure of angle DBC equals 36 degrees.

The more challgening method for measuring angles using Algebra is to assign a degree measurement to one of the adjacent angles, and to also assign a degree measurement to the resultant angle. Using the second example from above: if I am given an obtuse angle, ABC, and I indicate that the angle is composed of two adjacent angles, ABE and EBC, then the measure of ABC = the measure of ABE plus the measure of EBC. I can assign the algebraic expression of (2x + 5) to angle ABE, and the measurement of 60 to angle EBC. I also assign the measure of 120 to angle ABC. Therefore, the measure of angle ABC : 120=(2x + 5) + 60. To solve for x, and to obtain the degree measurement of angle ABC, calculate the value of x.

120 = 2x + 5 + 60.

120 = 2x + 60

120 - 60 = 2x + 60- 60.

60 = 2x

x = 30.

Angle ABE therefore equals 2 (30) + 5 = 65 degrees.

A search on the internet will not provide many links to pages that use this method of measurement instruction. The student should turn to the appropriate pages of the textbook and complete sample problems in order to master the skill.

The Basics

For a quick review of the basics in Geometry, follow these links:

http://library.thinkquest.org/J002441F/index.htm

http://library.thinkquest.org/28586/index1024.html

Unpacking the Distance Formula

Lesson 12

Unpacking the Distance Formula

by

Mr. White

Foundations of Mathematics Instuctor

Southern California

What is the Distance Formula?

The Distance Formula is derived from the Pythagorean Theorem, and is studied in Algebra and in the beginning of Plane Geometry. What the formula attempts to do is to allow students to calculate the distance between two points that line on an oblique line within a coordinate plane. Most teachers and books do not stress the oblique line as part of their work to teach the Distance Formula. But, to me, why do we need the formula if not for oblique lines? Anyone can count the distance between two points in a coordinate plane if they have access to a coordinate plane to view the vertical or original line, true? We need the Distance formula only when the line segment being measured is neither horizontal or vertical.

One of the confusing parts of the distance formula for students is the formula itself. The formula appears below:

Ö(x₂-x₁)²+ (y₂-y₁)²

What Do the Parts of the Distance Formula Really Mean?

The Distance Formula means nothing if we do not understand its components and how they are used in the formula. We must remember that we are evaluating the distance between two points, (x₂, y₂) and (x₁, y₁). The first part of the formula, (x₂-x₁)² allows us to subtract the two x coordinates of the two ordered pair we are using in the equation. The second part of the formula,(y₂-y₂)² , allows us to subtract the two y- coordinates of this same ordered pair. But why do we perform a squaring operation when we subtract the x's and the y's? The squaring is necessary because we want the difference between the two x- and the two y- coordinates to be positive. Distances can never be negative. When we add (x₂-x₁)²+ (y₂-y₁)², we are combining the value of the squared differences of the x's with the value of the squared differences of the y's. These squared distances are positive. We then take the quare root of this sum to obtain the true distance. Taking the square root of this squared number gives us the root( or the true value) for the distance between the two points.

Using the Distanceformulaa in an Example

Let us now use the paragraph above to guide us through a problem using the distance formula.

Given two points, A (0 , 0) and B ( 4, 3 ), what is the distance between the two points?
We first set up our equation:
Ö(4-0)²+(3-0)²
We solve that part of the equation involving the x- coordinates. 4-0 = 4. 4² = 16.
We then solve that part of the equation involving the y-coordinates. 3 - 0 = 3. 3² = 9.
We now have Ö(16 + 9) . This equals Ö(25). From Algebra we know that the square root of any positive number a is ±a. Since distance can never be negative, we chose the positive square root, 5. Therefore, the distance from point A to point B is 5 units.

Summary and Useful Web Links

All of the problems you will encounter using the Distance Forumla involve math facts that are described above. Use the links to the web pages below for additional leaopportunitiesnities.

http://whyslopes.com/Calculus-Introduction/distance_formula_and_midpoint_ca.html

http://www.purplemath.com/modules/distform.htm

The Midpoint Forumla Made Easy

Lesson 11 - Understanding the midpoint formula.
by
Mr. White
Foundationsl of Mathematics Instructor
Southern California

Vocabulary -

Midpoint
Coordinate/ ordered pair
X-
Y-
Line Segment
Endpoints
Average
Coordinate Geometry

The Midpoint Formula is used in coordinate geometry to allow us to determine the middle coordinate of a line segment. This point is also called the line bisector. It is as if you were measuring a line segment with a ruler. The middle of the line segment would be the midpoint.

For example, given two points, A (0,0) and B (4,3), what is the midpoint of the line that connects the two points, line segment AB? We could draw the line on a coordinate graph, but that is not necessary. We can use the Midpoint Formula, instead. This formula requires you to add the two X coordinates and average them. Then, add the two y coordinates, and average them. Those two answers are put together in the form of an ordered pair, which is the location of the line segment's middle point.

A (0, 0) B ( 4 , 4).

Calculate the midpoint of segment AB:

(4 + 0) /2 , (4 + 0)/2

This leads to the next step:

4/2, 4/2


which is the midpoint,

( 2, 2 )


Try another example.

Given the two points, R (1,2) and P (6, 5), what is the midpoint of the line segment, PR?

Solution: Add the x- coordinates, and divide by two, for the x-coordinate of the midpoint. Add the y- coordinates, and divide by two, for the y-coordinate of the midpoint.

Calculate the midpoint:

(1 + 7) / 2 , (2 + 6) / 2


This leads to the next step:

8/2, 8/2

or

(4, 4)

For more examples, go to these websites:

http://www.purplemath.com/modules/midpoint.htm

http://cs.selu.edu/~rbyrd/math/midpoint/

http://teacherschoice.com.au/Maths_Library/Analytical%20Geometry/Alg_16.htm

http://www.ronblond.com/M10/mid/index.html (A very interesting applet)