Archive for June 2013
THE SUM OF THE ANGLES OF A TRIANGLE (Part 2: Exploration)
This is the second part of the proof that the sum of the measures of the angles of a triangle is 180 degrees. This is actually the second method. The first one is more on using manipulatives or visual representations. This time, let us use basic mathematical concepts in proving.
This method is applicable to any type of triangle.
Let us use only one of the triangles. The process will be the same for the other triangles. To start with, let us name the triangle as ABC.
Now, let us draw a line parallel to base AC passing through B. Let us name this line as line BD or line BE or line ED, in any way you want it.
Since the focus of this proof is on the angles, let us rename each angle using numbers. It will be easier for us to determine the angles using the numbers instead of using three letters.
In this case,
Since we have drawn a line parallel to line AC, then we could say that side AB and side side BC are transversals of the parallel lines BD and line AC. Let us recall the concept of alternating interior angles for parallel lines.
Since we know that these alternate interior angles are always equal, then in the figure that we have formed, angle 1 = angle 4 and angle 3 = angle 5.
You comments ad suggestions are welcome here. Write them down in the comment box below. Thank you!
THE SUM OF THE ANGLES OF A TRIANGLE (Part 1: Exploration)
Everybody knows that the sum of the measures of the angles of any triangle is 180 degrees. If I may ask each one of you why, one of the reasons that I may probably hear is that "Our math teacher told us!" - which should not be the case. You should know how to show that the sum of the angles of any triangle is really 180 degrees. Where do 180 degrees come from? Why 180 degrees? Why not 360 degrees?
The purpose of this post is to show you how to prove that the sum of the angles of any triangle is really 180 degrees. This is the first method, which is the elementary way - the easiest way, to prove it. The other methods will also be posted here.
PROVE: The sum of the measures of the angles of any triangle is 180 degrees.
PRE-REQUISITE: The sum of the measures of the angles that form a straight line is 180 degrees.
MATERIALS NEEDED: colored papers or cardboards, pair of scissors, ruler, marker
PROCEDURES:
1. Cut three different types of triangles, classified according to angles. Use the colored papers or cardboards for the triangles. One should be a right triangle, another should be an acute triangle and the last should be an obtuse triangle.
2. For us to easily identify the angles of the triangles later, highlight the edges of each triangle using a black (or any dark colored) marker.
CONCLUSION: The sum of the measures of the angles of any triangle is 180 degrees.
For educators and parents, I have made a worksheet for this activity for your class/children. You are free to download and print it. You may group your students and let each group work on a triangle or all the triangles. I hope this will be helpful.
1 = 2?
Maybe you are wondering why or how can 1 be equal to 2. Let us look at the following proof:
First, let us have two real numbers a and b where
Using the addition property of equality, let us add b on both sides.
If we simplify both sides by combining like terms, it will become
Now let us subtract 2a on both sides
Combining like terms, we will arrive at
The right side has a common factor, which is 2. Using distributive property, it can be rewritten as
If you notice, both sides has a common factor, which is b - a. To simplify the equation, let us divide both sides by the common factor.
The process will arrive at
Are you convinced? No no no...
Seems like the proof is valid but there is something wrong with it. Look over the proof once again. Can you identify which of the process is not valid?
There is nothing wrong with the given. Real numbers can be equal. There is nothing wrong also with adding b and subtracting 2a on both sides. Likewise, there is nothing wrong with combining like terms on both sides. Then, where is the mistake?
There is nothing wrong with dividing both sides by any number but in this case it becomes invalid. The reason is that b - a = 0 since a = b. Subtracting equal numbers will yield 0. Since b - a = 0, then the result will be UNDEFINED. We cannot also cancel out b - a on both sides because of that.
Therefore, 1 is not equal to 2.
Here is a copy of the proof in pdf form. You may download and print it for educational purposes. You may share it to your friends. Your comments and suggestions are also welcome here.
