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Glencoe Geometry: Online Textbook Help13 chapters | 152 lessons

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Lesson Transcript

Instructor:
*Elizabeth Foster*

Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.

Watch this lesson to learn about two inequality theorems that can help you compare two triangles to each other. You'll also go through some practice problems with complete explanations.

When you're trying to figure out the side lengths and angle measures of a triangle, you aren't just limited to looking at other pieces of that one triangle. Sometimes, you can also use pieces of another triangle to help you figure it out. In this lesson, you'll practice two ways to do that, using two theorems about inequalities between two triangles.

The first theorem is the **SAS Inequality Theorem**, or **Hinge Theorem**. The SAS Inequality Theorem helps you figure out one angle of a triangle if you know about the sides that touch it. The theorem states that if two sides of triangle A are congruent to two sides of triangle B, and the angle between them is bigger in triangle A, then the third side of triangle A is bigger. If the angle between them is smaller in triangle A, the third side of triangle A is smaller. The SAS Inequality Theorem is basically an expansion of the SAS Theorem for proving two triangles are congruent. Where the SAS Congruence Theorem tells you about the cases where the side, angle, and side are all the same, the SAS Inequality Theorem tells you about the cases where the angles are different.

The second theorem is the SSS Inequality Theorem. The SSS Inequality Theorem states the converse of the SAS. If two sides of triangle A are congruent to two sides of triangle B, and the third side of triangle A is bigger, then the angle between the first two sides is also bigger in triangle A. The SSS Inequality Theorem is similar to the SSS Proof of Congruence. Proving congruence means proving that all three sides are the same; this theorem covers what you can know when one of them is different.

It sounds pretty complicated and hard to wrap your head around, but don't worry; its bark is worse than its bite. Once you start working on problems, it gets a little easier.

We'll start with a simple practice problem just to help you get all the sides and angles straight in your head.

*Side X is congruent to side Q. Side Y is congruent to side R. If side S is smaller than side Z, then angle B must be smaller than…*

This is a pretty straightforward test of SSS Theorem. Here, we have two pairs of congruent sides, each with an angle in between those sides. We know that the third side of each triangle is different: side S is smaller than side Z. So, by the SSS Inequality Theorem, angle B must be smaller than angle A, or smaller than 90 degrees.

Now let's look at something a little more involved.

*Line 1 and line 2 are perpendicular. Side RS and side SU are congruent. Side QS and side ST are congruent. Side TU must be greater than… *

*(a) Side QS(b) Side RQ(c) Side RS*

Let's break this down. From the one angle labeled 40 degrees, we know that angle RSQ = 40, because when two lines make an X shape, the opposite angles are the same. We also know that line 1 and line 2 intersect at 90-degree angles. So 40 + angle TSU = 90, which means angle TSU = 50.

Now we can use the SAS Inequality Theorem to solve this problem. The problem tells us that RS and SU are congruent, and that QS and ST are congruent. By the SAS Inequality Theorem, since angle UST is greater than angle RSQ, then side UT must be longer than side RQ.

Next problem!

*Sides X and W are congruent. Side V is longer than side Y. If the sum of the angle measures of angles A and B is 88 degrees, what is the largest possible integer value of angle A?*

In these two triangles, we have two pairs of congruent sides, each with an angle in between them. Side X is congruent to side W, and side Z is obviously congruent to itself. So by the SSS Theorem, we know that if V is greater than Y, then angle B must be greater than angle A.

The problem also states that A + B = 88. Let's think about all the pairs of numbers that add up to 88, where one is smaller than the other. We could have A = 1 and B = 87. Or, A = 5 and B = 83. Or, A = 20 and B = 68. But as soon as we hit A = 44, we run into trouble. If A = 44 B = 44, and that doesn't work because we know B has to be larger than A. So A always has to be smaller than 44. The biggest possible integer value would be A = 43 with B =45.

In this lesson, you learned about two theorems. Both of them help you compare two triangles. According to the **SSA Theorem** or **Hinge Theorem**, if two sides of triangle A are each congruent to the corresponding side of triangle B, and the angle in between those two sides is greater in triangle A, then the third side of triangle A will be longer than the corresponding side of triangle B.

The **SSS Theorem** is related. It states that if two sides of triangle A are congruent to the corresponding sides of triangle B, and the third side of triangle A is longer, then the angle in between the congruent sides of triangle A will be bigger than the corresponding angle in triangle B. That's a lot of words, but it starts making more sense once you put it into practice.

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Glencoe Geometry: Online Textbook Help13 chapters | 152 lessons

- Median, Altitude, and Angle Bisectors of a Triangle 4:50
- Perpendicular Bisector Theorem: Proof and Example 6:41
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- Inequality Theorems for Two Triangles 5:44
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