SSS and SAS

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

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.

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

Image 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;

$\overline{MX}=\overline{XN}\overline{MY}=\overline{YN}$

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

Therefore;

$△YMX\cong △YNX$

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

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

An 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.;

$△MON\cong △PQR$

Then;

$Areaof△MON=60m^{2}Areaof△PQR=60m^2$

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

$Areaof△PQR==\overline{PR}\times \overline{QR}\times \cos (\angle PRQ)60m^2=10m\times \overline{QR}\times \cos 60°$

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

$\overline{QR}=\frac{60m^2}{10m\times \cos 60°}$

Remember that

$\cos 60°=0.5$

Therefore,

$\overline{QR}=\frac{60m^2}{10m\times 0.5}\overline{QR}=\frac{60m^2}{5m}\overline{QR}=\frac{{\cancel{60}}^m\cancel{m^2}}{\cancel{5^1}\cancel{m}}\overline{QR}=12m$