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