PYTHAGOREAN THEOREM (Exploration 1)
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| The Pythagoras' Theorem |
The theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the remaining sides. That is c^2 = a^2 + b^2, where c is the hypotenuse, a and b are the lengths of the remaining sides.
Here is one of the theorem. Take note that the theorem only applies to right triangles.
Start with a cutout of any right triangle.
Hope this will help you understand the derivation of the Pythagoras' Theorem.
Your comments and suggestions are welcome here. You may write them in the comment box below. Thank you!
EXTERIOR ANGLE THEOREM (Part 2: Proof)
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EXTERIOR ANGLE THEOREM (Part 1: Exploration)
On the other hand, how many are you are familiar with the exterior angle theorem? What is the relationship of the exterior angles with the interior angles? The answer for these questions can be summarized by the "Exterior Angle Theorem".
In this post, let us discover where this theorem come from by using some illustrative examples.
Materials needed: ruler, protractor, calculator and pencil
Procedure:
1) On a piece of paper, draw one (1)
acute triangle, one (1) right triangle, and one (1) obtuse triangle. Name each
triangle as triangle ABC, with AC as the base.
2) Extend the base of AC to a point
D. Points A, C and D should be on the same line and C is between A and D.
- angle BCD is the exterior angle of each triangle
- angles A and B are the remote (non-adjacent) interior angles of each triangle
4) Look at the rightmost part of the table. Compare the sum of the measures of angles A and B and the exterior angle.
5) We can now make a conclusion/generalization about the relationship of the measurement of the exterior angle and the remote interior angles.
Here is the pdf copy of the activity, if to be conducted in a classroom setting. You are free to download it and use for your class. Hope it will be beneficial for you.
Your comments and suggestions are accepted here. Just write them in the comment box below. Thank you!
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.
