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Post by joey kimble on November 12, 2012

This is so simple, but for some reason my teacher makes is seem so complicated.

Inequalities Involving Two Triangles

  • SAS (Side-Angle-Side) Inequality Theorem (Hinge Theorem): If two sides of one triangle are congruent to two sides of another triangle, and the included angle in one triangle is greater than the included angle in the other, then the third side of the first triangle is longer than the third side in the second triangle
  • SSS (Side-Side-Side) Inequality Theorem: If two sides of one triangle are congruent to two sides of another triangle, and the third side in one triangle is longer than the third side in the other, then the angle between the pair of congruent sides in the first triangle is greater than the corresponding angle in the second triangle

Inequalities Involving Two Triangles

Determine whether the following statement is true or false.

If DB ≅ AB , then CD > AC.
True.
Determine whether the following statement is true or false.

If AB ≅ DE , BC ≅ EF and m∠A > m∠D, then AC > DF.
False.

AC ≅ BD , AB > CD.
Write an inequality comparing the angles, ∠ACB and ∠DBC.
  • AC ≅ BD , BC ≅ BC , and AB > CD
so m∠ACB > m∠DBC.
Determine whether the following statement is true or false.
If two sides of one triangle are congruent to two sides of another triangle, and one angle in one triangle is greater than one angle in the other triangle,
then the third side of the first triangle is longer than the third side in the second triangle.
False.

BC ≅ EF , AC ≅ DF , m∠ACB > m∠DFE, AB = 3x + 8, DE = 2x + 9
write an inequality for x.
  • AB > DE
  • 3x + 8 > 2x + 9
x > − 1.

BC ≅ EF , AC ≅ DF , AB = 8, DE = 9,m∠ACB = 5x + 6, m∠DFE = 12,
write an inequality for x.
  • AB = 8, DE = 9
  • AB < DE
  • m∠ACB < m∠DFE
  • 5x + 6 < 12
x < [6/5].

Isosceles triangle ABC, AB ≅ BC . Determine whether the following statement is true or false.
If m∠ADC > m∠BDC, then AD > BD.
False.

Isosceles triangle ABC, AB ≅ BC . Determine whether the following statement is true or false.
If AD > BD, then m∠ACD > m∠BCD.
True.
Fill in the blank in the statement with sometimes, never or always.
If two sides of one triangle are congruent to two sides of another triangle and the third side in one triangle is longer thanthe third side in the other,
then the angles between the pair of congruent sides in the first triangle is ____ greater than the corresponding angle inthe scond triangle.
Always.

Write a two - column proof.
Given: AF ≅ EF , FC⊥BD, BC ≅ CD , m∠AFB > m∠DFE
Prove: AB > DE.
  • Statements; Reasons
  • FC⊥BD, BC ≅ CD ; Given
  • FC ≅ FC; Reflexive prop ( = )
  • ∆ BFC ≅ ∆ DFC ; Right triangle LL
  • BF ≅ DF ; Definition of congruent triangles
  • AF ≅ EF , m∠AFB > m∠DFE ; Given
  • AB > DE; SAS Inequality theorem
S
tatements; Reasons
FC⊥BD, BC ≅ CD ; Given
FC ≅ FC; Reflexive prop ( = )
∆ BFC ≅ ∆ DFC ; Right triangle LL
BF ≅ DF ; Definition of congruent triangles
AF ≅ EF , m∠AFB > m∠DFE ; Given
AB > DE; SAS Inequality theorem

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Inequalities Involving Two Triangles

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • SAS Inequality Theorem 0:06
    • SAS Inequality Theorem & Example
  • SSS Inequality Theorem 4:33
    • SSS Inequality Theorem & Example
  • Extra Example 1: Write an Inequality Comparing the Segments 6:08
  • Extra Example 2: Determine if the Statement is True 9:52
  • Extra Example 3: Write an Inequality for x 14:20
  • Extra Example 4: Two-column Proof 17:44

Transcription: Inequalities Involving Two Triangles

Welcome back to Educator.com.0000

The next lesson is on inequalities involving two triangles.0002

Now, the last lesson that we did was on the triangle inequality theorem, which said that0007

any two sides of a triangle had to be greater than the third side; but that was within just one triangle.0015

We are going to go over a couple of inequality theorems that are based on comparing two triangles.0020

The first one is the SAS inequality theorem; now, SAS sounds familiar, right?--that should sound familiar.0027

We used the SAS theorem to prove that two triangles were congruent.0036

So then, we had to prove, within two triangles, that a side, an angle, and another side, were congruent.0042

And then, we could prove that these two triangles were congruent, and that was the SAS theorem.0053

The SAS inequality theorem is a little bit different; "inequality" means that it is not equal.0060

So then, we are not proving that two triangles are congruent; you would use the SAS theorem for that.0069

But the SAS inequality part of it is actually a little bit different.0077

The SAS inequality theorem is actually also known as the Hinge Theorem.0084

And it says, "If two sides of one triangle are congruent to two sides of another triangle, and the included angle0095

(the included angle was the angle between the two sides--it can't be angle A or angle C; it has to be angle B,0104

the angle between the two sides) in one triangle is greater than (not congruent to) the corresponding included angle0111

of the other triangle, then that side opposite, the third side, is going to be greater than the side opposite the other triangle."0124

There is a little twist to it; so for this one, the side and the other side are congruent, just like in the SAS theorem.0136

But the angles, the included angles of both triangles, are not congruent; one is actually going to be greater than the other.0144

In this case, let's say the measure of angle B is greater than the measure of angle E; this is the included angle of this triangle,0155

and this is the included angle of this triangle; so the first condition is that the two sides have to be congruent;0167

the second condition is that the included angle is either less than or greater than the other included angle.0173

Then, the side opposite...if this, then...AC is greater than DF.0182

That is the SAS inequality theorem.0205

Now, if the measure of angle B was less than the measure of angle E, then the side opposite AC, the third side, would be smaller than DF.0208

That is the SAS inequality theorem; now, don't get confused between the two.0222

You can either use the SAS theorem to prove that those two triangles are congruent by Side-Angle-Side,0227

or you can use the SAS inequality theorem if you are trying to say that one side is greater or less than the other side.0235

If this is, let's say, 10, then this has to be 10; if this is, let's say, 12, then this has to be 12.0247

And if this is 60 degrees, and this is 50 degrees, then we can use the SAS inequality theorem to say that AC is then going to be greater than DF.0259

The second one, the SSS inequality theorem, is the same exact thing, but you are going the opposite way now.0276

It is the same thing, where you have two sides being congruent to two sides; so this S and this S are still the same.0286

They have to be congruent; but then, when it comes to the third side, again, it is not congruent.0295

With the third side, you are now comparing the two; and of course, this is after the two sides being congruent.0301

If AC is greater than DF, then the angle opposite the measure of angle B, the greater side,0308

is going to be greater than the angle opposite the shorter side.0325

It is almost the same as the SAS inequality theorem; but then this time, you are actually using the sides now.0333

You are using the sides, and then you are making a conclusion about the included angles.0341

With the SAS inequality theorem, you used the included angles to compare, and then based that on the side opposite.0346

With the SSS inequality theorem, you are using the sides to compare, and then basing that same comparison on the angles.0356

We are just going to go ahead and work on our example problems.0370

Write an inequality comparing the segments.0373

We are going to compare AB with CD.0377

Now, you can see the two triangles right here: we have triangle ABD, and we have another triangle DCB.0381

We are going to compare this side, AB, with CD.0392

Now, for us to use the SAS or SSS inequality theorem, we have to first know that, no matter what,0397

in both triangles, the two triangles that we are comparing, two sides have to be congruent to two sides of another triangle.0421

And then, it is either the A or the S that we are using the inequality for.0428

So, for AD, I know that this is 20; and then BC--I know that that is 20; so then, they are automatically congruent.0435

And then, BD is shared between both triangles; so then, I can just do that.0443

So, if you can see this: now I have two sides of a triangle being congruent to two sides of the other triangle.0453

So, side congruent; side congruent; that is one of the sides; then this side is congruent to this side of the other triangle,0464

so that is the other side; and then, the included angle, we are comparing.0475

So, if you can see that this is a triangle, and then this is a triangle; two sides are congruent to these two sides;0484

then, this angle compared to that angle...I know that I can use the SAS inequality theorem.0492

Then, using that, I am going to make a conclusion based on these sides opposite, this one with that side.0501

Then, the included angle of this triangle, the measure of angle D, is less than the measure of angle...0512

well, maybe I should not say the measure of angle D; I need to say that if the measure of angle ADB,0524

or I could say BDA, is less than the measure of angle CBD, then I can conclude that AB, the side opposite,0534

is less than...and then, what is the side opposite this one?--CD.0552

AB is going to be smaller than CD, and that is SAS.0557

Again, side and side must be congruent--the same thing here: this side and that side must be congruent.0564

And then, based on the included angle or the other side (you are going to have inequalities comparing those two),0569

either one, then you can make the conclusion about the angles opposite or the sides opposite.0577

Some of them used this to make a conclusion about the sides opposite.0585

OK, with CD as a median, determine if the statement is true.0591

If you remember a median of a triangle, remember that the endpoint is from a vertex all the way down to the side opposite,0599

so that it is hitting the midpoint; think of median as "midpoint" or "middle."0608

The m gives that away: midpoint or median or middle.0616

Since we know that CD is a median, I know that this AB is cut into two equal parts.0622

That means that AD is congruent to BD; this and this are congruent.0630

And then, since I need two triangles to compare, because of SAS and SSS inequality theorems,0638

I need two triangles; so then, here are my triangles; that is triangle ACD and triangle CBD.0659

I have one side taken care of; that is either this one or this one.0671

And then, this side right here--if they share a side, then that is automatically congruent by the reflexive property.0676

So now, I have Side (this is the included angle), Side, and this is Side, included Angle, Side.0684

Now, we have both sides being congruent.0693

Now, for this first one, if the measure of angle 1 is less than, is smaller than the measure of angle 2, then BC,0699

which is this one right here, is greater than AC.0711

This can be a little bit confusing, because it is switched around.0720

If the measure of angle 1 is smaller than the measure of angle 2, well, let's flip this around, then.0726

If these are flipped around, then let's flip this around.0732

If the measure of angle 1 is less than the measure of angle 2, so this one is smaller than this one,0737

then I can say that the measure of angle 2 is bigger, or greater, than the measure of angle 1.0744

This and this are the same thing; so now, I can say that the measure of angle 2 is greater than the measure of angle 1.0751

Then, BC, which is the side opposite (this is BC) is greater than the side opposite angle 1 (is AC).0759

So, is that true? That is true.0773

So again, the measure of angle 1 is smaller; this one is smaller, and this one is bigger.0780

That means that BC, the side opposite, is bigger than AC, which is smaller.0788

And then, that second one: if AC (let me just use a different color)--this is #2--is greater (this is bigger now)0799

than BC (this is the smaller one), then the measure of angle 1 is smaller than the measure of angle 2?0822

No, because if this is bigger, then this angle has to be also bigger.0836

If this is smaller, then this one has to be smaller.0840

The fact that they said that AC is greater than BC, but the measure of angle 1 is smaller than the measure of angle 2, means this one is false.0845

The next example: we are going to write an inequality for x.0862

Let's take a moment to look at the diagram and see what we have here.0867

Well, I see two triangles; that is good, because I need two triangles to use the two inequality theorems.0875

And I see that this triangle, one of my triangles, is an equilateral triangle.0883

And then, I see that this triangle is an isosceles triangle.0890

I need to compare my x's; now, my x here has to do with this side; my x here has to do with this side.0897

So, what can I conclude about these sides (again, with SAS and SSS)?0911

I need two of my sides to be congruent, so what two sides do I have to work with?0921

I have this one and then...I have these two, because I have to conclude about the side opposite.0929

That means that I need to use this included angle.0942

And then, the same thing here happens for this side, and then this side, something with that angle, and then something with this angle.0945

I have to conclude about this and about this to make it based on this side.0960

If this is 40, then do we have two sides being congruent to two sides of the other triangle?0969

Yes, we do, because here is my one side, and here is the side of the other triangle; so that is the side.0976

Then, I have this side of this triangle, with this side of the other triangle--whichever theorem I am going to use.0985

Then, I need to look at my angles to see how I can use my included angles to make a conclusion about the sides opposite.0997

If this is 40, do I know what this angle measure is?1009

Well, if it is equilateral, then it is also equiangular; that means that each angle is 60 degrees.1013

This included angle is greater than this included angle; that means that this side opposite is going to be greater than this side opposite.1022

And then, I can just simplify this; so if I subtract the x, I get x + 8 is greater than 10.1038

Subtract the 8; x is greater than 2; so there is my inequality for x.1051

And then, for my fourth example, I am going to write a two-column proof, and let's take a look at our given:1065

AB is congruent to DC, so those two are congruent; and then, AB is also parallel to DC, so I can write those symbols there.1072

The measure of angle BEC is greater than the measure of angle CED, so this is the bigger, and this is the smaller, angle.1088

And then, I want to prove that BC is greater than CD.1105

Well, since I know that this angle is greater than this angle, the side opposite is going to be greater than the side opposite this angle.1112

But before we do that, I need to know...because all I have here is this; this is all of the information that I have,1124

because that is the angle; the angle is bigger than the other angle, because I am trying to compare BC, this one, with this one;1137

and if this angle opposite is bigger than this angle opposite, then this side has to be greater than this side;1149

but I have to first know that this side and this side have to be congruent, because I need two sides1160

to be congruent to two sides of the other triangle, and then you can base1169

this whole bigger and smaller thing with the angles and the sides opposite.1174

I need to say, first of all, because this is the included angle...I need to know that these two sides of this triangle1181

are going to be congruent to these two sides of this triangle.1189

That way, it will be SAS, and this angle will be included for this triangle, and then this will be the included angle for this triangle.1194

So, how do I do that? Well, I have to basically prove...1203

Now, I know that I am trying to prove that these two are going to be congruent to these two.1211

Well, I know that these are already congruent to each other, because of the reflexive property.1218

Now, I have to prove that BE, the other leg, or the other side, is going to be congruent to this side.1233

Now, there is no way to actually prove that those sides are congruent, unless I first prove that maybe these two triangles,1242

this triangle and this triangle, are congruent; then I can say, "Well, then, this side is going to be congruent to this side, because of CPCTC."1257

And again, why am I doing that?--because I need to prove that this side and this side are congruent.1268

It is based on these two triangles; but then it is these two sides.1281

This side is already congruent to each other, so I already have one side covered.1289

This side is good; but now I have to show that this side, the other side of this triangle, is congruent to the other side of this triangle, which is this.1292

So, statements/reasons: my first step is to say that AB is congruent to DC; AB is parallel to DC.1303

And then, the measure of angle BC is greater than the measure of angle CD; the reason is "given."1329

#2: I can say that EC is congruent to EC, because of the reflexive property.1339

And then, there is my side; that is one of the sides there.1362

And then, I want to say...now I am going to try to prove that the triangles are congruent, these two triangles,1369

so that I could say that these two corresponding sides are congruent by CPCTC.1378

Since I know that BA is congruent to CD, that is one side.1386

So, here, this is one side; and this is different than this side.1392

This side is for the SAS inequality theorem; but for us to show that these two triangles are congruent, I am going to use this side;1402

then I am going to use, let's see...now that they are parallel, I know that alternate interior angles are congruent;1414

so then, I know that this angle and this angle are congruent, so angle BAE is congruent to angle...if I use BAE, then I have to use DCE.1422

And then, my reason; and if you can't really see that, these are the parallel lines; that is AB and CD.1444

And then, there is the transversal; I am saying that this angle right here is congruent to this angle right here.1454

And that is this angle and this angle.1462

If two parallel lines are cut by a transversal, then alternate interior angles are congruent.1467

So then, this is a side done and an angle done, and then I know that these angles are also congruent,1491

so angle BEA is congruent to angle DEC, and the reason for that is "vertical angles are congruent."1505

There is another angle; so now, I can say that my triangles are congruent; triangle ABE is congruent to triangle CDE.1527

And I have to make sure that it is corresponding; ADE is corresponding to CDE.1548

And what was the reason for that? Angle-Angle-Side.1555

And the whole point of proving that these two triangles are congruent is just so we can have this side being congruent.1567

That is the only reason why we are proving that these two triangles are congruent,1578

because to prove that this S is congruent takes this and this, and the only way we can prove them congruent1582

is if we just prove that these two triangles are congruent, and then we say that corresponding parts of those triangles are congruent.1591

So then, now I can say that BE is congruent to DE; my reason is CPCTC.1596

Remember: you can only use this once you prove that the triangles are congruent.1617

Then, you can say that any corresponding part is congruent.1620

If you ever have to prove that two parts--maybe two sides or two angles--are congruent,1624

then you have the option of doing that--proving that the triangles are congruent, and then1629

saying that those corresponding parts are congruent by reason of CPCTC, Corresponding Parts of Congruent Triangles are Congruent.1634

Now, we are done with this one; we have this--we are good with this--and we are good with the sides.1643

Now, remember: we said that this angle is bigger than this angle; this is the bigger angle; this is the smaller angle.1651

Now that we have all of our information, now we can say that BC, then, is greater than CD.1658

And what was the reason? SAS inequality theorem.1669

So again, this whole step right here, #1, 3, and then 4--the whole reason why we had to use that, and step 5,1677

to prove that the triangles are congruent, is so that we can say that one of the sides is congruent.1691

Otherwise, if these two sides are not congruent, we can't use the SAS inequality theorem.1696

Even if we know that the angle is bigger than the other angle, and so the side opposite must be bigger than the other side,1700

it works only if the two sides are congruent--that is why we had to prove all of that, so that we can use CPCTC.1706

So, if you are still a little lost or confused with this problem, I want you to just watch the slide again.1717

All that we had to do--I know it looks like a lot of work, but a big chunk of it was just to prove this pair of corresponding sides congruent,1727

because we had this by the reflexive property; we have one of the sides being congruent;1740

and we didn't have the other pair of sides being congruent.1746

So then, that is why we had to prove the triangles congruent.1749

And then, we can say, "OK, those sides are now congruent by CPCTC."1752

We had this information; we had this information; and then, all of that work was to have this third piece of information.1757

And then, once you have all of that, you can say your conclusion, based on the SAS inequality theorem.1762

That is it for this lesson; thank you for watching Educator.com.1773