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.