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### Course: Geometric Measurement & Modeling 221-230 > Unit 2

Lesson 3: Inequalities in a triangle# Ordering triangle sides and angles example

Sal first solves a problem where he orders the sides of a triangle given the angles, then solves a problem where he orders the angles of a triangle given the sides.

## Want to join the conversation?

- Can you use the angle to figure out how long it actually is?(18 votes)
- Not if you only know the three angles, you need at least one side. You can see this for yourself: draw a triangle on a piece of paper, it doesn't matter which angles you pick. Outside the triangle, next to each of the sides, draw another line parallel to the triangle's side. If you do this for all three sides, you'll get a second triangle which is bigger than the original, but has exactly the same angles. From that painting you can see that there is more than one triangle with exactly the same angles, but one is bigger than the other. In fact, there are infinitely many of such triangles!

Once you know one side, you can use the law of sines to find the others. In case you're interested, here is the law of sines:

a / sin(A) = b / sin(B) = c / sin(C)

Where a is the length of one side and sin(A) the sine of the angle across from side a (and similar for b, B, c, and C).(6 votes)

- Can we get accurate length of third side with the help of two sides(3 votes)
- No, we can't, because although the length of the third side depends on the lengths of the other two sides it also depends on the angle between the two sides.

However, if we had the two lengths and the angle, we could find the third side by using the law of cosines, which you can watch an example of here:

https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-law-of-cosines/v/law-of-cosines-example

I don't know how familiar you are with the trigonometric functions, but the cosine function is one of them and they have in common that they make use of various relations between angles and sides in right triangles.

If it's completely new to you, you can watch an introduction to trigonometry here:

https://www.khanacademy.org/math/trigonometry/trigonometry-right-triangles/intro-to-the-trig-ratios/v/basic-trigonometry(6 votes)

- 0:44I don't get it. "The side that this angle opens up to is going to be the shortest side of the triangle." I'm really stumped. What is the concepts of (The angle that this opens up to) and how is it always going to be the shortest side of the triangle if there's three?(6 votes)
- It's because the angle, 57 degrees, is the smallest of the three angles.

If you try increasing the angle measure, you'll notice that the opposite side will increase in length to compensate for the wider angle. Likewise, decreasing the angle will decrease the opposite side's length.(0 votes)

- What is a good way to remember the order?(4 votes)
- At0:54sal says " the next
**largest**angle"

But I think he meant to say " the next**smallest**angle"(4 votes) - why are some triangles so confusing sometimes.(4 votes)
- in the first video he say they have given the interior angels of the triangle what that mean?(3 votes)
- The angle measures on the inside of the triangle. Hope this helps!(2 votes)

- can u tell me some tip how to order the smallest to largest?

and from largest to smallest?

i just need some clue please everyone!(2 votes)- I found it very well explained in the video.

Really, what he's saying is that with only angles and not side lengths for any given triangle, the smallest interior angle (the one on the inside of the triangle) will have the largest once directly on the opposite side of the triangle.(3 votes)

- are there more than just the common angles like acute, obtuse,and right?(2 votes)
- Yes, there are many different types of angles, besides those three. There are straight, reflex,vertical, opposite, corresponding and 360 degree angles (just to name a few)(3 votes)

- so if I just type in some numbers they would turn blu

0:78(2 votes)- The blue number is nothing more than the time on the video. 0:78 is impossible as after 60 seconds it turns into1:01.(2 votes)

## Video transcript

- [Voiceover] We're asked
to order the side lengths of the triangle from shortest to longest. And we have the three sides here, and we could use this little tool to order them in some way. If we look at the triangle, we've been given the interior
angles of the triangle, and they haven't told us
the actual side lengths. So, how are we supposed
to actually order them from shortest to longest? Well, the realization
that you need to make here is that the order of the lengths of the sides of a triangle are related to the order
of the measures of angles that open up onto those sides. What do I mean by that? Well, let's think about these
three angles right over here. 57 degrees, that is the
smallest of these three, and so the side that
this angle opens up to, or you can think of it
as the opposite side, is going to be the shortest
side of the triangle. So, b is going to be the shortest side. So, the next largest angle is 58 degrees, and so a is going to be the middle side. It's not going to be the
longest nor the shortest. So, a. Then 65 degrees, that
opens up onto side c, or the opposite side of that angle is c. So, c is going to be the longest side. To get an intuition for why that is, imagine a world where the 65 degree angle, if we were to make it bigger. If we were to make the
65 degree angle bigger, maybe by moving this point out and that point out, what would happen? Well, side c would get bigger, and because the angles of a triangle have to add up to 180 degrees, if this one's getting bigger, these will have to get smaller. Likewise, if I were to take angle... let's say, if I were to
take this 58 degree angle, and if I were to make it smaller, what's going to happen? Well, side a is going to get smaller. So, the general principle, I'm not giving you any formal proof here, but the intuition is, is that the order of
the angles will tell you what the order of the
sides are going to be. So, the smallest side is going to be opposite the smallest angle. The largest side is going to
be opposite the largest angle. We can check our answer,
make sure we got it right. Now, let's do one that
goes the other way around. Here, we want to order
the angles of the triangle from smallest to largest, and we're given the sides. Well, same, exact idea. The smallest angle is going to be opposite the smallest side or the shortest side. Well, the shortest side is
this side of length 7.2. The angle that opens
up onto it is angle a. So, that's going to be the smallest angle. Then, the next smallest side
is the side of length 7.3, and the angle that opens up to it is angle b right over here. So, angle b. Then angle c opens up
onto the largest side. So, it's going to be the largest angle. So, we're done. We've ordered the angles of the triangle from smallest to largest.