The Angle Business and Beginning Geometry
By
Mr. White, Foundations of Mathematics Teacher
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.
