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Triangles - Part I


Now we can start talking about triangles. A triangle is a shape formed by three line segment sides. And of course, triangles are important. First of all, there are a lot of important facts about triangles themselves, and also we can understand many other shapes by thinking of them as triangles put together.

So, they play a very important role in geometry. Let's talk about some basic terms for triangles. The line segments are called sides, obviously. Each corner where the two line segments meet, where there's an angle, we call that a vertex. And the plural of that word is vertices.

So, a triangle has three vertices. In fact, every triangle has exactly three sides and exactly three vertices. This video lesson concerns some of the basic facts that are true for all triangles, every single triangle on Earth. In later videos, we'll start talking about some special triangles, but this video's really about all triangles.

The most important and most remarkable, is that for any triangle, no matter what the individual angles are, the sum of the three angles inside any triangle on Earth is exactly 180 degrees. That is an absolutely remarkable fact proved first by Mr. Euclid, and it is 100% true for all triangles. So, why is this true?

Why would that remarkable fact be true? Well, think about it this way. Consider the following diagram. Triangle ABC could be any triangle. Line DE is drawn so that it is parallel to AC. So, remember what we talked about in the last video, when we have parallel lines, and a transversal cutting across those parallel lines.

Well, if we consider AB a transversal, then those two red angles, BAC and ABE. Those two red angles are equal. Those are congruent angles there. If we consider BC a transversal, then the two blue angles, ACB and CBD, those are also congruent. And of course, whatever that yellow angle is, in the middle, we can see that if we look along the line, those three angles add up to 180 degrees because they, they lie along a line, red, yellow, blue, those three angles.

And so it must mean that the three angles in the triangle also add up because it's the same angles, they also add up to 180 degrees. So that's why this works for any triangle. Understanding this may help you remember that fact and understand it a little more deeply. Now some helpful terms for angles.

First of all, an acute angle is an angle less than 90 degrees. A right angle, of course, is 90 degrees and an obtuse angle is more than 90 degrees. We'll be using these terms throughout our discussion of geometry and it's possible that they could appear on the test. Because the sum of the angles in a triangle is always 180 degrees, a triangle cannot possibly have two right angles or two obtuse angles.

At least two of the angles in any triangle have to be acute. A triangle could have one obtuse angle and two acute angles. So we have a diagram here, and incidentally, all the diagrams in these lesson videos are drawn to scale. So that is a scale diagram of a triangle with 20, 120 and 40 degrees. Triangle could have one right angle and two acute angles.

Here's a 30, 60, 90 triangle. Or a triangle could have three acute angles. So 50, 60, and 70 degrees. In a triangle, the angles at any vertex is between two sides, it touches those two sides, or you could say it's adjacent to those two sides.

For example, angle ABC, the angle right here, touches side AB and side BC. It's adjacent to those two sides. The third side, the side that the angle does not touch, is opposite the angle. For example, that same angle ABC and side AC are opposite one another. So that's an important relationship and we'll be talking about that a lot, the relationship of the angle and its opposite side.

Now we can talk about another rule true for all triangles. The largest angle is always opposite the longest side, 100% of the time. The largest angle's opposite the longest side, and the smallest angle is always opposite the shortest side. In the triangle above, of course drawn to scale, AB is clearly the longest side. So that means that angle C has to be the largest angle.

And in fact, angle C looks like it's obtuse. An obtuse angle in a triangle is always gonna be the largest angle. BC is clearly the shortest side, and so that means that angle A must be the smallest angle. So we absolutely know that this is always true. The final rule that is true for all triangles is called the Triangle Inequality Theorem.

The Triangle Inequality Theorem states, in any triangle, the sum of any two sides must be greater than the third side. Notice this is a strict inequality. It's not greater than or equal to, but strictly greater than. Why is this true? Well, think about a triangle.

Here's a typical triangle. Clearly, the shortest distance between two points has to be the straight line path. That's just, that's a fundamental geometric fact. The shortest distance between two points is always a straight line. Well, that means the shortest possible distance between point A and point C must be the segment AC, because AC is a straight path between A and C.

The detour path from A to B to C has to be longer than AC, because AC by definition is the shortest possible path. Therefore, it must be true that that detour path, AB plus BC, has to be longer than AC. And we can say the same analogous thing for the other pairs of sides, that AC plus BC has to be bigger than AB by itself, and AB plus AC has to be bigger than BC by itself.

Another way to think about the triangle inequality. Suppose the sum of two sides was less than the third side. Suppose we tried to construct say, a triangle with lengths 2, 3, and 10. So here's a scale diagram, and you can kinda see there's a problem here. Because no matter how we move that side of a 2 and a side of 3 we can rotate it all we want, they're not gonna touch.

No matter where we move the short arms, they are simply not gonna reach across that gap to connect. So we're not actually gonna be able to join it up and close off the shape as a triangle. So this does not form an legitimate triangle. It has to be closed.

The three line segments all have to meet and enclose an area in order for it to be a triangle. So this will not work as a triangle. And this is why it's impossible to have two sides in a triangle that sum to less than the third side.

What if the sum of the two sides equals the third side? Remember, the equals is not allowed by the triangle inequality either, it's a strict greater than. So what's the problem with equals? Well, what happens if we try and construct a triangle with sides 4, 6, and 10? So 4 plus 6 equals 10.

What's going on here? Well, you see the problem here. I mean, yes, we can get the, the side of 6 and the side of 4 to connect up with each other, but the problem is, everything is flat. And, what we have, there's not really a geometric word for it, so I'm gonna call it a flat thing.

That flat thing is not a triangle. A triangle has to have an inside. It has to have area. So this flat thing does not meet our requirement for what a triangle is. This is what happens if we have two sides equal. Their sum is equal to the, to the third side.

And so that, this is why not only the less than case is excluded, but also the equal to case is excluded. The sum of two sides has to be greater, and only greater, than the third side. Suppose we know that two sides of a triangle are 8 and 13, and we want to know the possible lengths of the third side.

Well, certainly it's true that if we take the sum of the unknown side plus 8 it has to be bigger than 13. Well from this inequality, we could subtract 8 from both sides and it means that that x has to be greater than 5. So that's one of the statements we have. And notice that 5 is the difference of 8 and 13.

It's also true that the sum of 8 and 13 must be greater than that third side. So it means that x has to be less than 21. Notice that's the sum of the two sides. So, we can say that x must be greater than 5 and less than 21. That's the allowable range for x. And, in particular, notice that it is less, it is greater than the difference of the sides and less than the sum of the sides.

Well we can generalize this. If we know that the lengths of two of the three sides of a triangle, we know that they're P and Q, then the third side has to be greater than the difference and less than the sum. Any side of a triangle must be more than the difference of the other two sides, and less than the sum of the other two sides.

So this is an alternate way to see the triangle inequality. In summary, the sum of three angles in any triangle must be 180 degrees, that's the really big one, true for all triangles. Two angles of a triangle must be acute. The third one could be acute, right or obtuse. But, it's true for any triangle that at least two of the angles are acute.

Another property true for all triangles, the biggest angle is always opposite the largest side, and the smallest angle is always opposite the shortest side, true for all triangles. The sum of any two sides of a triangle is greater than the third side, true for all triangles. And another way to say that is any side of a triangle must be greater than the difference of the other two sides and less than the sum of the other two sides.

Everything on this summary is true for every triangle on Earth.

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