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SSS and SAS

SSS and SAS

Have you thought of making your life easier? Who hasn’t, at least on some occasions, right? Lots of times making your life easier just means boiling an action down to fast applicable, easy to remember techniques and learning it.

In this case – shortcuts to tell if two or more triangles are congruent or not. For this reason, five shortcuts or theorems have been introduced and abbreviated, which makes them easier to remember: SSS, SAS, HL, ASA and AAS. In this article, only the first two will be explained. More on the rest in the another article.

If you can spot only one of the following conditions between two or more triangles, then this means that the triangles are congruent to each other.

What is SSS?

Side-Side-Side or SSS for short is pretty simple to understand.

SSS theorem means that all three corresponding sides are equal between two or more triangles, which means that all corresponding angles are therefore equal too.

So if you spot SSS, think of congruency.

Let’s look at some SSS examples.

Two equilateral triangles are positioned next to each other. If you don’t know what is an equilateral triangle, this simply means a triangle with all equal sides.

Are these two triangles congruent?

Two equilateral triangles - StudySmarter Original

So, by mere observation, it can be seen that both equilaterals are indeed the same (equal) in both length and angle.

This is a great case to apply the SSS theorem to prove congruence. So, Side-Side-Side – all three respective sides are equal between these triangles. You can look at the picture above to understand this better. We can say that the first triangle is congruent to the second triangle:

Triangle_1 ≅ Triangle_2

The example above is a simple case because you don’t even need to look if the given sides are equal correspondingly, that is, whether the sides of one triangle are equal to the respective sides of the other triangle.

Why is this important? You can position the triangles in any way relative to each other and it will still be easy to tell that they are congruent, because all sides have the same length.

So let’s look at a case where it’s important to take into account the order in which we compare the sides of one triangle and the other in the next example.

Here are two triangles positioned differently from one another - one triangle is rotated relative to the other. Are these triangles congruent?

Differently oriented triangles - StudySmarter Original

By looking at the lengths of the sides of both triangles it is evident that the given triangles are congruent given the theorem SSS:

ABC ≅ DEF

In this example, the order of the letters also shows that the sides of one triangle are equal to the respective sides of the other triangle. ABC and DEF are arranged alphabetically and the respective sides AB, BC, CA are equal in length to DE, EF, FD consecutively. It’s not so evident when looking at the picture above, though. If there were no letters naming the triangles, you’d need to first understand that the triangles are rotated relative to each other.

Keep in mind that this may not always be the case as in the example above – in some cases, the names of the triangles are not matched alphabetically or arranged logically, but the triangles can still be congruent. Always look at how the triangles are positioned relative to one another first. The names of the triangles are arbitrary.

In the figure below, the line XY is equidistant to the line MN. Is the triangle YMX congruent with triangle YNX?

SSS and SAS Image of congruent triangles formed from equidistant lines StudySmarterImage 1 of congruent triangles formed from equidistant lines - StudySmarter Original

Solution

The line XY being equidistant to line MN means it cuts MN at its midpoint. This implies that;

SSS and SAS Image 2 of congruent triangles formed from equidistant lines StudySmarter

Image 2 of congruent triangles formed from equidistant lines - StudySmarter Original

Now, both triangles YMX and YNX have the same third side XY.

SSS and SAS Image 3 of congruent triangles formed from equidistant lines StudySmarter

Image 3 of congruent triangles formed from equidistant lines - StudySmarter Original

Therefore;

Let’s move on to the next theorem called SAS.

What is SAS?

Side-Angle-Side or SAS for short means that two corresponding sides together with the joining angle are equal between two or more triangles.

SAS is true because the length of the third side is pre-determined if the length of the remaining two sides and the angle they form is known. If two or more triangles have two equal sides with the same exact angle in between them, this means the given triangles are congruent.

Let’s look at a couple of examples of SAS.

Two triangles are given next to each other. The first triangle has one angle of 60º and the two sides forming it have both a length of 6. Same case with the second triangle. Are these triangles congruent?

Triangles with equal angles and respective sides - StudySmarter Original

You may think this is pretty easy, huh? Pretty trivial maybe?

That’s right! Just by looking at the picture, you can tell this is the same case as the first example of SSS, only the sides are of different lengths. In this case, however, the given info on the triangles is only of the lengths of two sides and the angle in between. If you already know equilateral triangles well, you can tell they are both congruent right away even without the picture.

If taking into account only the given info, the triangles in this example are congruent given the condition SAS:

Triangle_1 ≅ Triangle_2

Let’s try a bit more complex case.

Three triangles are positioned differently from each other. See the picture below.

Differently positioned triangles - StudySmarter Original

Are these triangles congruent?

You can see that the triangles are all rotated relative to each other. By looking at the given values on the triangles, we can see that ABC is not congruent to DEF because the angles between the corresponding equal sides AB, BC and DE, FE are not equal. However, ABC and XYZ are congruent due to the theorem SAS, because they both have equal respective sides and the angle formed by them is also the same:

ABC ≅ XYZ

Remember that triangle names are arbitrary and in some cases, the names of the triangles are not matched alphabetically or arranged logically. This is the case in the example above, but ABC and XYZ are still congruent due to SAS.

Let's go to further examples.

Let us go through an example to understand better what SAS and SSS mean as well as to observe the distinction between both.

SSS and SAS An image showing three diagrams of that portrays the SSS and SAS theorems StudySmarterAn image showing three diagrams of that portrays the SSS and SAS theorems - StudySmarter Original

The figure below consists of three diagrams labelled I, II and III. Determine the following:

a) Are they all congruent?

b) Which is (are) SSS congruent?

c) Which is (are) SAS congruent?

d) If the area of the ΔMON is 60m2 , ∠PRQ is 60° and line PR is 10m find QR.

Solution

a) From the figure above, diagram I have both triangles joined together have two of their sides and angle equal. Thus, with respect to the SAS theorem, we can say both triangles in I are congruent.

In diagram II, all three sides of both angles are the same; thus, in line with the SSS theorem, both triangles in diagram II are congruent.

In diagram III, both triangles have two of their sides and angle equal. Thus, with respect to the SAS theorem, we can say both triangles in III are congruent.

b) Based on the earlier solution from question a), we can say that only diagram II is SSS congruent.

c) Based on the earlier solution from question a), we can say that both diagrams I and III are SAS congruent.

d) Since triangles MON and PQR are SAS congruent, i.e.;

Then;

To find line QR when PR is given, we know that;

Make line QR the subject of the formula by dividing both sides of the equation by the product of 10m and cos60° to get;

Remember that

Therefore,

SSS and SAS - Key takeaways

  • There are five theorems for triangle congruence, which help to evaluate whether given triangles are congruent.
  • These theorems are SSS, SAS, HL, ASA and AAS;
  • SSS (Side-Side-Side) states that two or more triangles are congruent if all of their respective sides are equal;

  • SAS (Side-Angle-Side) states that two or more triangles are congruent if two consecutive sides are equal to that of another triangle and the respective sides form the same exact angle.

Frequently Asked Questions about SSS and SAS

SSS and SAS are theorems for proving triangle congruence. SSS and SAS are both abbreviations for Side-Side-Side and Side-Angle-Side, respectively.

The theory for SSS is the following - if two or more triangles have all corresponding sides of equal length, then the given triangles are congruent. Regarding SAS - if two or more triangles have two equal corresponding sides and the same angle in between, then the given triangles are congruent.

The SSS and SAS congruence theorems are theorems for proving congruence between two or more triangles.

SSS proves congruence between two or more triangles if all of the given triangles have equal corresponding sides.

To prove congruence using the SAS theorem, you need to find out the lengths of two consecutive sides and the angle between them. Once this is done, you need to compare the corresponding sides and angle to those of another desired triangle. If they are equal, then the given triangles are congruent.

Final SSS and SAS Quiz

Question

What is SSS?

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Answer

SSS is a theorem for proving congruency between two or more triangles. SSS itself refers to an abbreviation Side-Side-Side, meaning if all three sides are equal in length between two or more triangles, then the given triangles are congruent.

Show question

Question

What is SAS?

Show answer

Answer

SAS is a theorem for proving congruency between two or more triangles. SAS itself refers to an abbreviation Side-Angle-Side, meaning if two respective sides and the angle they form are equal between two or more triangles, then the given triangles are congruent.

Show question

Question

When do you use SSS and SAS?

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Answer

To prove if the given triangles are congruent

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Question

Three triangles are given. The sides of the first triangle are all equal to the respective sides of the second triangle. Two sides and the angle in between of the third triangle are equal to the respective two sides and angle of the second triangle. 

Check all the statements that are correct:

Show answer

Answer

All three triangles are congruent to each other

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Question

Are two equilateral triangles equal to each other if they both have a side of length 5 cm?

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Answer

Yes, this is because of the condition SSS.

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Question

Two right triangles are given. Both legs of the first right triangle are equal to those of the second triangle. Check all the statements that are correct:

Show answer

Answer

Both triangles are congruent

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Question

SAS and SSS are both theorems for proving triangle congruence. These theorems can be used on:

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Answer

Equilateral triangles

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Question

Can SSS and SAS be used to prove congruency between more than two triangles?

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Answer

Yes

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Question

Two triangles are given. 

First triangle has one side of length 5 cm and the other of length 7 cm. They both form an angle of 40o.

The second triangle also has two sides of the same length forming a 40o angle.

Are these triangles congruent? If so, which theorem can be used to prove congruency?


Show answer

Answer

The given triangles are congruent, this can be proved using SAS

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Question

Two triangles are given. 

First triangle has one side of length 5 cm and the other of length 4 cm. They both form an angle of 60o.

The second triangle also has two sides of the same length forming a 61o angle.

Are these triangles congruent? If so, which theorem can be used to prove congruency?

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Answer

The given triangles aren't congruent, because no theorem can be used for proving this

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Question

Three triangles are given. 

First triangle has one side of length 4 cm and the other of length 3 cm. They both form an angle of 90o.

The second triangle also has two sides of the same length forming a 90o angle.

The third triangle has sides of lengths 4 cm, 3 cm and 5cm

Are these triangles congruent? If so, which theorem can be used to prove congruency?

Show answer

Answer

These triangles are congruent. It can be proved by both SAS and SSS, though you'll need to calculate the third side of the first or second triangle to prove this. You can do this by using the Pythagoras theorem.  

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Question

If you prove that two triangles are congruent using SSS, does this mean they'll also have equal areas?

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Answer

Yes

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Question

If you prove that two triangles are congruent using SAS, does this mean they'll also have equal perimeters?

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Answer

Yes

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Question

Does using SSS and SAS mean you won't need to make any measurements to prove congruency?

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Answer

It can mean that, but you'll need to see if there is enough information given on the triangles.

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Question

If you're given two triangles and both have two equal respective sides, does this mean that these triangles are congruent?

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Answer

Not necessarily

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