Two triangles are congruent if one can be moved (by translation, rotation, and/or reflection) to lie exactly on top of the other. It is a more precise way of saying that two triangles are "the same".
Here is an example of two congruent triangles:
If we reflected one of these triangles, rotated it, and translated it, we could place it directly on top of the other.
When deciding whether or not two triangles are congruent, we don't need to know six pieces of information (three sides and three angles), we need only three pieces in one of these combinations:
If two triangles have three equal side lengths, then the triangles must be congruent. Try this yourself with three straight objects - once you put them together, you can rotate, translate, and reflect the triangle to make every other possible combination:
This kind of congruence is called side-side-side, or SSS.
If two triangles have a pair of matching sides and the angles between them are equal, then the triangles must be congruent. Try this yourself with two straight objects - if you hold them together at one end and form an angle, there is only one triangle you can form by joining the ends together:
This kind of congruence is called side-angle-side, or SAS. We write this test with the "A" in between the two "S"s, because the angle must be between the matching sides - the long name for this kind of congruence is "two sides and the included angle".
It is possible for triangles to have two pairs of equal sides and a pair of matching angles, yet not be congruent overall. Here is an example:
This can only happen when the pair of equal angles is not included between the sides.
Try using this applet to find the two different triangles that have two matching angles and a matching non-included angle, just like the picture above:
If two right-angled triangles have equal lengths hypotenuses and another pair of equal sides, then the triangles must be congruent:
Notice that the right angle is not included between the sides - this is the only exception to the general rule, which is why we mention it as a separate case. This congruence test is called right angle-hypotenuse-side, or RHS.
What if we are only given one pair of equal sides? In this case we need two pairs of equal angles. Here is the construction if the two angles are made with the given side:
If one of the given angles is opposite the given side, we can always find the third one by using the angle sum of a triangle:
This kind of congruence is called angle-angle-side congruence, or AAS.
To use AAS to show that two triangles are congruent, the matching sides must have the same position relative to the matching angles.
The three triangles below have two pairs of equal angles and a pair of equal sides, but because the side of length $7$7 is in a different position, none of them are congruent:
These are the four congruence tests:
If two triangles don't satisfy these tests, one of two things could be true:
Two triangles are definitely not congruent if there is a pair of sides or angles that are in the same relative position but are not equal to each other.
Are these triangles congruent?
Think: Both have matching sides with lengths $9$9 and $5$5. We can use the angle sum of a triangle to find the missing angle in the first triangle.
Do: In the first triangle, we subtract the known angles to find the missing angle: $180^\circ-56^\circ-44^\circ=80^\circ$180°−56°−44°=80°. We now know that the first triangle doesn't have a right angle, so it cannot be congruent to the second triangle.
Are these triangles congruent?
Think: Both have matching sides with lengths $7$7 and $6$6. We don't know anything about the angles in the second triangle. They may look congruent, but that isn't enough - either we know for sure, or we don't.
Do: There is no way to calculate any additional side or angle information in the second triangle, and we need at least three matching pairs to make conclusions about congruence. We don't know whether or not they are congruent.
Consider the two triangles in the diagram below:
Are $\triangle PQR$△PQR and $\triangle STU$△STU congruent?
Yes, they satisfy SSS.
Yes, they satisfy SAS.
Yes, they satisfy AAS.
Yes, they satisfy RHS.
No, they are definitely not congruent.
Unknown, there is not enough information.
Consider the following:
Which two of the following triangles are congruent?
What congruence test does this pair satisfy?
SSS
SAS
AAS
HL
Consider the following diagram:
Are the triangles $\triangle ABD$△ABD and $\triangle CDB$△CDB definitely congruent?
Yes
No
What congruence test does this pair satisfy?
SSS
SAS
AAS
RHS
Select the three statements that, when put together, establish congruence for this test.
Make sure each reason is correct as well.
$\angle ADB=\angle CBD$∠ADB=∠CBD
Alternate angles on parallel lines
$AD=CB$AD=CB
Corresponding sides on parallel lines
$\angle ABD=\angle CDB$∠ABD=∠CDB
Corresponding angles on parallel lines
$AB=CD$AB=CD
Given
$BD$BD is common
$\angle ABD=\angle CDB$∠ABD=∠CDB
Alternate angles on parallel lines