Friday, June 28, 2013

mathematics,geometry,angles,triangles,sum of angles,IGCSE

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. 
mathematics,triangles,angles,geometry,right triangles,obtuse triangle,acute triangles,IGCSE
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.
mathematics,angles,triangles,geometry,equilateral triangle,IGCSE
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.
mathematics,geometry,angles,triangles,interior angles,exterior angles,parallel lines,acute triangles,equilateral triangles,IGCSE
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.
mathematics,geometry,angles,triangles,interior angles,exterior angles,parallel lines,IGCSE
In this case, 
mathematics,geometry,angles,triangles,IGCSE

Let us take note that the angles now of the triangle are angle 1, angle 2 and angle 3.

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.
mathematics,geometry,angles,triangles,parallel lines,equal angles,IGCSEmathematics,geometry,angles,triangles,parallel line,transversal line,IGCSE

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.
mathematics,geometry,angles,triangles,acute angles,interior angles,exterior angles,IGCSE
If you notice, angles 2, 4 and 5 form a straight line. Let us recall
mathematics,geometry,angles,straight lines,supplementary angles,180 degrees,IGCSE
That means the sum of angles 4, 2 and 5 is 180 degrees, because they form a straight line.
mathematics,geometry,angles,exterior angles,triangles,interior angles,supplementary angles,IGCSE
From the illustrations above, let us recall that angle 1 = angle 4 and angle 3 = angle 5.
mathematics,angles,geometry,alternate interior angles,triangles,IGCSE
Further, it means that the sum of angles 1, 2 and 3 is also 180 degrees.
mathematics,geometry,angles,triangles,sum of the angles,IGCSE
Therefore, we can conclude that
mathematics,geometry,angles,triangles,sum of angles,IGCSE
You can also use the other sides of the triangle for the proof. The same process will be used in each of the cases.
mathematics,geometry,triangles,angles,IGCSE
mathematics,geometry,angles,triangles,IGCSE
Here is the summary of the proof in pdf form. You may download and print for academic use. Hope it will become useful to you.
You comments ad suggestions are welcome here. Write them down in the comment box below. Thank you!
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