Geometry Inequalities for Sides and Angles of a Triangle

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The next lesson, we are going to look at some more inequalities and inequalities comparing sides and angles of a triangle.0002

To look at sides to angles, we are going to compare; and this theorem says that, if one side of a triangle0012

is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side.0023

And again, these theorems are within one triangle; it is not comparing a triangle to another triangle,0034

which we have been doing in the past few lessons.0040

Within a triangle, if one side is longer--let's say BC is longer; let's say this is 20, and it is longer than another side--0045

so another side, let's say, is 15; then the angle opposite the longer side (then the angle opposite BC would be angle A)--0058

that means that since BC is greater than AB, the angle opposite the longer, which is angle A,0070

is going to have a greater measure than the angle opposite this side, which is angle C.0082

That is what it is saying; that means that, since BC is greater than AB, the measure of angle A is going to be greater than the measure of angle C.0093

So, if a side is greater than another side, then the angle opposite that greater side is going to be greater than the angle opposite the shorter side.0106

And the same is true for its converse; so again, if an angle measure is greater...let's say angle A0118

has a greater angle measure than angle B; so let's say this is 60 degrees, and this is 50 degrees;0129

then the side opposite the greater angle, BC, is going to be greater than the side opposite the smaller one, so greater than AC.0138

Now, this is a very short lesson; we are just going to go over some examples.0159

We are going to name the angles in order from least to greatest.0165

I know that AC is the shortest side, of 6; then it is AB, of 8, and then it is BC, of 10.0170

So, for the sides, we have this being the shortest, AC; then AB; then BC as the greatest.0178

Then we can just name the angles in that order; and of course, it has to be the angle opposite.0189

So, angle B is going to be the least; then it is going to be (let me write the measure of angle B)--the second is AB,0195

and the angle opposite that is angle C, so the measure of angle C; and then, the greatest angle is going to be0211

the angle opposite the greatest side, which is the measure of angle A.0219

And the next example is the same thing, but now we are going to name the sides from longest to shortest.0227

Here, this is the greatest angle; remember, since we are going from longest to shortest,0239

and the greatest angle is C, the side opposite that is going to be the longest, then, so that is AB.0246

Then, since we are talking about measure, let's not write that segment; it is only for congruence if you have that little line above it, the segment bar.0254

The next one is angle B; the side opposite angle B is AC.0266

And then, the shortest side is going to be the side opposite the smallest angle measure, which is 38.0274

So, the side opposite that would be BC.0284

Example 3: Find the value of x and list the angles and the sides of triangle ABC in order from shortest to longest.0292

So, if you have a triangle ABC, and here they are giving us all three angle measures...0298

and they want us to just list the angles and the sides in order from least to greatest or shortest to longest...0316

So, I know that the measure of angle A, plus the measure of angle B, plus the measure of angle C, is equal to 180.0324

And that is the angle sum theorem: all three angles of a triangle add up to 180.0340

So then, I am just going to substitute all of this in: 9x + 18 + 22 + 11x + 10x - 10 is going to equal 180.0345

Let's see, I am going to add up my like terms, all of the x's.0363

9x + 11x is 20x, plus 10x is 30x; and then, 18, 22, and -10...18 + 22 is 40, minus 10 is 30; so 30x + 30 = 180.0369

I am going to subtract 30; so 30x = 150; and then, when I divide by 30, x is going to equal 5.0393

That is not the end of the problem: we found the value of x, and then we have to list the angles and sides in order from shortest to longest.0409

Before I can start on my sides, I have to know what my angle measures are.0421

I have to plug x back in to find the measure of angle A; the measure of angle A is 9(5) + 18, which is 45 + 18; that is 63 degrees.0425

The measure of angle B: 22 + 11(5) is going to be 77 degrees.0445

And then, the measure of angle C is 10(5) - 10; that is 50; that is 40 degrees.0456

Let's just double-check our work to see that all of this is going to add up to 180.0469

63 + 77 + 40...that is 10...10...80...so that is right; all three add up to 180, so we have the correct angle measures.0474

Then, you can just use any diagram; just draw triangle ABC, so that you can see what sides are opposite the angles.0489

Angles in order from least to greatest would be the measure of angle C (that is the smallest),0502

and then the measure of angle A, and then the measure of angle B (has the greatest angle); that is for the angles.0513

And then, for the sides: the side opposite the measure of angle C is going to be AB;0525

for A, it is going to be BC; and then, for B, it is going to be AC.0534

That is Example 3; and the last example is going to be a proof.0545

Here, let's see what we have: BD bisects angle ABC (remember: "bisect" means it cuts it in half,0554

so that means that BD cuts this angle in half; that means, since it cuts it in half, that now I have two congruent parts).0565

And then, I want to prove that the measure of angle ADB, this angle right here, is greater than the measure of angle ABD.0578

So then, this angle is greater than this angle right here; I am comparing these two angles.0592

Well, let's see what we have to work with: now, this angle right here, I know, is going to be greater than this angle, which is what I have to prove.0601

I know that this angle is going to be greater than this angle right here.0620

And these are the same; so then, that means that this has to be greater than this.0627

How do I know that this angle is greater than this angle?0630

If I draw...OK, well, that is not a very good picture; it doesn't look anything like that; let me draw that again...something like that,0635

where this is D, this is B, and this is C, then all I did was draw this part right here (minus this--I didn't draw this part).0652

That means that we can see that this is an exterior angle.0662

So, if this is angle 1, then my two remote interior angles are going to be this one and this one.0666

And in the last section, we went over the exterior angle inequality theorem.0673

And the exterior angle inequality theorem says that the exterior angle is going to be greater than either of its remote interior angles,0682

because the exterior angle theorem--not the inequality theorem, just the theorem itself--0694

says that the measure of angle 1 is equal to the measure of angle 2, plus the measure of angle 3.0699

These two together add up to the measure of angle 1.0707

But the exterior angle inequality theorem says that the measure of angle 1 is going to be greater than each of these by itself.0712

So, this is what I am going to use to say that this angle is greater than this angle.0724

And then, I can say, "Well, this angle and this angle are the same; therefore, this has to be greater than that angle."0733

Statement and reason: #1: BD bisects angle ABC, and that is given.0742

Since it bisects the angle, I can say that angle ABD is congruent to angle DBC.0772

And the reason for that is just "definition of angle bisector."0799

Then, the third one: I am going to change this from "congruent" to "equal," because here, I want to use the measures,0813

since my "prove" statement, my last statement, says "measure of angles."0829

Here, this is congruency; so I can just switch over, just so this step will lead to this step.0834

All I am going to do is just measure of angle ABD = measure of angle DBC, and then "definition of congruency."0843

You can say "definition of congruency"; you can say "definition of congruent angles."0857

And then, I am going to use this theorem to say that the measure of angle ADB is greater than the measure of angle DBC.0868

Again, that is just the measure of angle 1 being greater than the measure of angle 2.0891

It is the same thing; and that is the exterior angle inequality theorem.0895

And then, from there, I can say that since this is greater than this, I am going to say that now this is greater than this.0913

So, the measure of angle ADB is greater than the measure of angle ABD.0924

So then, since this is now the same as this, if I label this as 4, they are congruent.0940

So then, I can just say that the measure of angle 1 is then greater than the measure of angle 4, since these two are the same.0950

And that is all I did; and then, that would be the substitution property.0959

Again, just to explain how I did the proof here: to get from this step to this step, BD bisects this angle, ABC;0975

that means it is cut into two equal parts, and that is just the definition of angle bisector.0985

You can see that each of these angles is congruent to each other (definition of angle bisector).0989

Then, from this step to this step, I just changed it from congruency to equality (definition of congruency).0994

And then, from there, we used the exterior angle inequality theorem, where the measure of angle 11005

is going to be greater than each of these remote interior angles; so the measure of angle 1 is greater than the measure of angle 2.1016

And then, since this and this are the same, and that is written right here--DBC and ABD--1025

I just substitute this in for that, because that is what we have to prove.1036

That is it for this lesson; thank you for watching Educator.com.1043