Explorations into the Properties of Various Angles and Triangles
This essay focuses on the various properties of angles and triangles, taken from the perspective of a Middle School Math student. To begin this exploration a student must first familiarize themself with the various properties of angles.
What are some observations you can make about these three lines and their angles?
You can see that there are two parallel lines intersected by a transversal line and that line cuts through both the parallel lines at the same angle. Also there are two sets of vertical angles that are congruent with one another with the same angle measure. Another characteristic of these lines is that the angles on either side of the transversal line equal 180 degrees and angles on either side of the parallel lines equal 180 degrees. Also, alternate interior angles are congruent, alternate exterior angles are congruent and the corresponding angles are congruent. The following image shows the lines with the angle measures.
Middle School Mathematics is built upon the idea of students scaffolding their math knowledge to learn new concepts. Now that students have explored the ideas of corresponding angles and their measures, they can look more specifically at various angle measures with respect to their corresponding sides.
What observations could you make about two angles with corresponding perpendicular sides?
By the color coding of the two angles, you can see that the two "red" sides of the angles are perpendicular to one another and the two "blue" sides are perpendicular to one another. If you take the measure of the two angles, you can see that no matter what the length of the sides is, the measures will remain congruent. An animated GSP file will demonstrate the previous statement.
What observations can be made about two angles with corresponding parallel sides?
By the color coding of the two angles, you can see that the two "red" sides of the angles are parallel to one another and the two "blue" sides are parallel to one another. If you take the measure of the two angles, you can see that no matter what the length of the sides is, the measures will remain congruent. An animated GSP file will demonstrate the previous statement about parallel corresponding angles.Once students have a firm grasp on various angle properties, they can be introduced to Triangles!
What do we know about triangles?
After examining the basic triangle shape we can determine many different properties. A triangle has three sides and three interior angles, but what are some different questions we should ask ourselves? Do the lengths of the sides have anything to do with the angles? What should the interior angles total to? What are exterior angles? What should the exterior angles total to?
With this randomly constructed triangle we can see that each of the interior angle measures are different, but they always total to equal 180 degrees.
What about angle bisectors in a triangle?
When we look at the various properties of this triangle we can see that many things are going on. We see that each dotted line is an angle bisector of each interior angle of the triangle and they each intersect each other at one common point inside the triangle. What are some other questions we could ask ourselves about this figure? Is the center equi-distant from each side of the triangle? If we measure the angle bisectors we would see that the center point is relatively the same distance from each side because this is a randomly constructed triangle, but if the triangle were an equilateral triangle the measures would be more consistant.
What about other kinds of bisectors?
Here we constructed a triangle then found the perpendicular bisectors of each of the three sides and we know that no matter where the bisector crosses the side of the triangle, it is a 90deg angle because it is perpendicular. What else do we see? You can also see that the bisectors created another triangle within the original triangle. If we were to take the measures of both of the triangles we would find that they are similar triangles because each of their angle measures are the same. This GSP file shows the construction with the measurements.