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The measure of angle one will be greater than angle 2, and
The measure of angle 1 will be greater than angle 3 3
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1 # Triangle Inequality Theorems

​Transcript Triangle Inequality Theorem

Today we are going to look at triangle inequality, and there are actually two ways to look at triangle inequalities. The first way, say you are given the three angles or even two of the angles you can solve for the third one, and the other way is to have the three sides. So let’s look at the first method with the angles. We have the angles of forty; sixty, and eighty and we need to list our sides from smallest to greatest. We already know our smallest angle is forty degrees so I’m going to go to my smallest angle and I’m going to go across that triangle and that means that side B is my smallest side because it is opposite my smallest angle. So B is my smallest angle. Now let’s go to the next angle. So 60 is the next largest angle. What is it opposite of? It’s opposite of side A so side A is the medium side is eight is across from side C That means that side C is my largest angle. This is not drawn to scale so don’t be confused by my labeling. So in my diagram based on my labeling side B is my smallest, side A is my medium, and side C is my largest Now let’s look at it in reverse . Let’s look if we are given the three sides. You do the exact same thing except go back to the angle. So three is my smallest side and it is opposite angle B, so angle B is my smallest angle. What’s my next smallest angle? I go to my next smallest side, which is four. It is opposite angle A. So angle A is my medium sized angle, and my largest side is the side with 5 and it is opposite angle C or the right angle. So that is the order of the angles and here is the order of the sides. Let’s look at the rules. What are the rules for Triangle inequality? First you’re going to draw across the triangle to the opposite side or the opposite angle. I always start with my smallest, and you can label them smallest to largest then fill them in with an inequality symbol. So the smallest one is less than the middle sized one, which is less than the largest one. Hope this video was helpful.  C
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List the sides of Triangle ABC  from smallest to largest.

Side B is the smallest because it is across from the smallest angle.

Side A is the next largest because it is across from the next largest angle.

Side C is the largest side because it is across from the largest angle.
Triangle ABC
Triangle ABC
List the angles of Triangle ABC from smallest to largest.

Angle B is the smallest because it is across from the smallest leg.

Angle A is the next largest becausre it is across from the next largest leg.

Angle C is the largest side because it is across from the largest leg.
A triangle must meet these three triangle inequality theorems in order to be classified as a triangle. If these theorems related to triangles are NOT true, you do not have a triangle
Let's look at two sample problems Common Core Standard: 7.GA.2
If given two or three angles how do prove triangle inequality?

If given side length how do you prove triangle inequality?

If given three sides of a triangle, which angle has the smallest angle measure?

If given three angle measures, which side is the shortest? the longest?
Triangle Inequalities Conjecture

The longest side will be opposite the greatest angle measure,therefore, the shortest triangle side will be opposite the smallest angle measure.

The greatest angle measure will be opposite the longest side and the smallest angle measure will be opposite the smallest side.

The sum of any two sides of the triangle must be greater than the third side.

The angle measure of the exterior angle of a triangle is larger than the measure of either nonadjacent interior angle.