Proof – Sum of Interior Angles of a Triangle 128-2.11

Proof – Sum of Interior Angles of a Triangle 128-2.11


This video is provided as supplementary
material for courses taught at Howard Community
College and in this video I’m going to prove that
the sum of the interior angles of a triangle is
equal to 180 degrees. So I’m going to start out with a
demonstration, not yet a proof. I’ve got a piece of paper
that I cut into a triangle and I’ve labeled the three
angles A, B and C. What I’m going to do is rip the paper into three pieces so that one of the original angles is on
each piece. And then I’m going to put those three pieces together so the three angles all meet along a straight edge. And what
I can see is that where they meet, they form
what appears to be a straight line. So this is just a demonstration. Let’s do
an actual proof make sure that it works for all triangles.
So I’ll draw a triangle and I’ll label the three angles A,B and C. And then I’ll extend the base so it’s a line, and I’m going to draw a line parallel to
the base at the top of the triangle.
So now I have two parallel lines. I’m also going to extend each of the other sides and now each of those sides forms a transversal that cuts through the
parallel lines. Now remember, when you have a pair of parallel lines and a transversal, the alternate interior angles that are formed are equal angles. So for angle A, the alternate interior angle would
be up at the top here next to angle B. And I’ll just call that A-prime. So angle A and angle A-prime are equal. So angle A equals angle A-prime And for angle C, the alternate interior angle would be up at the top next to B.
Well call that C-prime. and angle C and angle C-prime are also equal. So angle C equals angle C-prime. Now looking at angles A-prime, B and C-prime, we notice that the form a straight line along that parallel line at the top the triangle. So angle A-prime plus angle B plus angle C-prime add up to 180 degrees, that straight line. I’m sorry, that should be C-prime. And now if we replace A-prime with A, you we can do that
because they’re equal, and add angle B, and replace C-prime with C, because
they’re equal, and add that to it, we find the angle A plus angle B plus angle C add up to 180 degrees. So there’s the proof that the three interior angles of a triangle add up to 180
degrees. Take care, I’ll see you next time.


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