Proof that a Triangle is 180 Degrees

The three angles of a triangle add to be 180°. There are many different ways to see this. One way is to simply add up the interior and exterior angles of the triangle.


Proof that a Triangle is 180 Degrees
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Proof that a Triangle is 180 Degrees


Proof that a triangle is 180 degrees. The three angles of a triangle add to be 180 degrees. There are many different ways that you can see this. One way is if you construct a triangle out of paper, make sure those lines are straight, and you tear off those three angles, then when you put them together they’ll make a straight line.


But, here’s another proof of why the three angles of a triangle add to be 180. Here we have two parallel lines. Line DE and line BC and that’s important. Our lines must be parallel in order for this proof to be true.


Since those lines are parallel then we can see that angle DAB and angle ABC are alternate interior angles. They’re on different sides of this transversal AB and there inside are parallel lines. We call those alternate interior angles.


We have a proof about alternate interior angles that says, “if two lines are parallel then alternate interior angles are congruent”. We know that these angles are congruent to each other, which is why I’ve marked them each with one arc.


Likewise, angle EAC and angle ACB are also alternate interior angles because they’re on different sides of this transversal and there inside our parallel lines. Again, those angles are congruent to each other because of the alternate interior angles theorem.


Then finally, angle BAC, if you put it together with these other two angles you get a straight line which measures 180 degrees. Since angle DAB, plus BAC, plus CAE, equals 180. Then because these angles, alternate interior angles are congruent, and these alternate interior angles are congruent, that means that the three angles inside of our triangle also add to be 180 degrees.



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Last updated: 10/04/2018

 

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