The two triangles shown are congruent. Circle the correct reason that proves that they are congruent.
Two triangles are congruent if they are exactly the same size and shape. They may be in different positions or orientations, but if we were to cut one out, we’d be able to place it exactly on top of the other. This means that the three angles in one triangle are the same as the three angles in the other. And also the three side lengths in the two triangles are the same.
However, we don’t have to prove that all three pairs of side lengths are the same and that all three pairs of angles are the same in order to prove that two triangles are congruent. We can just prove a subset of this information. There are four conditions that we can use to prove triangle congruency.
The first condition, SSS, stands for side side side. And we just need to prove that the two triangles have the same three side lengths.
The second condition, SAS, stands for side angle side. And here we need to prove that the two triangles have two sides and an included angle in common. It must be the included angle, so that’s the angle between the two sides whose lengths are the same in the two triangles. It’s not enough to show that two triangles have two sides and any angle in common.
The third option, ASA, stands for angle side angle. So here we need to show that two triangles have two angles and an included side in common. Now actually for this condition, it isn’t always essential that it’s the included side. We can actually show that two triangles have two angles and any side in common. However, as this statement has been written in the order angle side angle, if we want to use this condition to prove congruency of our two triangles, we must use two angles and an included side.
The final condition, RHS, is specific to right-angled triangles. R stands for right angle, H stands for hypotenuse, and S stands for side. If we can show that two right-angled triangles have the length of the hypotenuse the same and the length of either of the other two sides equal, then we’ve proved that they are congruent.
Let’s look at the information that we’ve been given for our two triangles. We can see that the two triangles have an angle of 71 degrees in common. They also have a side of three centimetres in common and an angle of 39 degrees in common, which suggests that it’s the angle side angle congruency condition that we need to use.
However, notice that the side of three centimetres is not the included side between the angles of 71 degrees and 39 degrees. Instead, it’s included between the angle of 71 degrees and the third angle in the triangle. However, we can work out the size of this angle as the angles in any triangle sum to 180 degrees. The sum of the two known angles of 71 degrees and 39 degrees is 110 degrees. And subtracting this from 180 gives 70 degrees.
Now we have our side of three centimetres included between the angles of 71 degrees and 70 degrees. So we can use the angle side angle condition to prove that the two triangles are congruent. This is why though it doesn’t actually matter in general whether the side is included or not, because if we know two angles in a triangle, we can always work out the third using the fact that angles in a triangle sum to 180 degrees.
So if we know any two angles and one side in a triangle, we’ll be able to use the angle side angle, or perhaps we could call it angle angle side or side angle angle, condition to prove that two triangles are congruent.