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Proofs for Triangles

Instructor: Sierra Clegg

Sierra has a Bachelor of Science in mathematics and a Master of Arts in Teaching. She has taught high school math for three years.

Triangles are perhaps the most important shape and have more theorems than any other polygon. In this lesson we'll explore just a few of those important facts about triangles and why those facts are true.

Three is a magic number

Past, present, and future. Faith, hope, and charity. There's no denying three is a magic number. In the mathematical world the number three leads us to think of the triangle. Yes every triangle has three sides, but more basic than that, they have three points, or vertices. Three lines can exist and not make a triangle (think parallel) but it is impossible to have three points that don't. So in honor of the magical three, this lesson is about proving some facts about it's magical partner- the triangle.

Theorems and Proofs

A theorem is a statement in math that we prove to be true. Theorems are not obvious facts like triangles have three sides. Simple definitions like that don't require proving, they just are. A theorem is a fact that we wouldn't know immediately if or why it's true. Instead, we have to show, or prove, it's true using some logic. Take the fact that all the interior angles of a triangle add up to 180o. This is a fact many students learn early on, but where did it come from? Why is it true? Let's prove it!

Triangle Sum Theorem

Take any old triangle and label it's angles 1, 2 and 3.

Triangle123

We can't assume anything about them, especially what we're trying to prove. This means we can't just assign them number values because that's too specific and proofs need to be generalized to work for every case. That's where reducing the problem to something simpler can help. One thing simpler than a triangle, is a line. Take our triangle and draw a line parallel to one side and through the opposite vertex like so:

Triangle12345

This creates two more angles we'll call 4 and 5. Angles 2, 4 and 5 all fit together on that new line and you may recall lines, or straight angles are also 180o. So we have three angles adding up to 180o, just not the three we want. The key to making this work is that the line we created is parallel to that side. When two parallel lines are both intersected by a third line (our other two sides of the triangle) it creates congruent angles known as alternate interior angles. Alternate interior angles are between a set of parallel lines but on opposite sides of that intersecting line such as angles 1 and 4 and angles 3 and 5. This means even though what don't know their exact measure, we know they are the same. So if angles 2, 4 and 5 add up to 180o we can just substitute angles 4 and 5 with 1 and 3 and now all our interior angles add up to 180o.

Isosceles Triangle Theorem

The great thing about the Triangle Sum Theorem and the way we proved it, is it's true for every triangle. We didn't use any facts about the triangle itself because we didn't know anything. For more specific triangles, we can use their properties to help us prove more about them. For example, an isosceles triangle is defined as having two sides that are congruent or the same. We can use this to prove the theorem that an isosceles triangle's base angles (the angles opposite the congruent sides) are also congruent.

Consider this isosceles triangle:

Isosceles

The tick marks indicate the sides are congruent and we are trying to prove angle 1 is congruent to angle 2. Once again, we're going to use an extra line to help us out and put it right down the middle of the triangle.

BisectedIsosceles

Because of the congruent sides we have, this line is a line of symmetry. This means every piece on the left of the line matches to a corresponding piece on the right. Thus, angle 1 matches and is congruent to angle 2.

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