Definition: Triangles are
when all corresponding sides and interior angles are
The triangles will have the same shape and size, but one may be a mirror image of the other.
In the simple case below, the two triangles PQR and LMN are
because every corresponding side has the same length, and every corresponding angle has the same measure.
The angle at P has the same measure (in degrees) as the angle at L, the side PQ is the same length as the side LM etc.
Drag any orange dot at P,Q,R. The other triangle LMN will change to remain congruent to it.
One triangle can be rotated, but as long as they are otherwise identical, the triangles are still congruent.
In the figure below, the triangle LMN is congruent to PQR even though it rotated anti-clockwise about 30°
In the figure below, drag any orange dot at P,Q, R and the other triangle will change to remain congruent to it.
One triangle can be a mirror image of the other, but the triangles can still be congruent if
the corresponding sides and angles have the same measure.
It can be reflected in any direction, up down, left, right or anything in between.
In the figure below, triangle LMN is a mirror image of PQR, but is still considered congruent to it
because the corresponding sides are congruent and the corresponding angles have the same measure.
The side QR corresponds to side MN and is congruent to it. The angle P corresponds to and has the same measure as the angle L , and so on.
Drag any orange dot at P,Q or R. The other triangle LMN will change to remain congruent to it.
They can have common parts
Congruent triangles can also have a common side or vertex that is shared by both triangles.
For example in the figure below, the two triangles have the side QR in common and are congruent.
Note that they are also reflections (mirror images) of each other.
Because the two triangles have QR in common, those two sides are of course equal, and so this fact can be used when trying to show they are congruent. Only two other pieces of information would be needed.
Try thisDrag any orange dot at P,Q and R.
The other triangle, which shares a side QR with it, will change to remain congruent to it.
Imagine the triangles are cardboard
One way to think about triangle congruence is to imagine they are made of cardboard.
They are congruent if you can slide them around, rotate them, and flip them over in various ways so they
make a pile where they exactly fit over each other.
How to tell if triangles are congruent
Any triangle is defined by six measures (three sides, three angles).
But you don't need to know all of them to show that two triangles are congruent.
Various groups of three will do. Triangles are congruent if:
- SSS (side side side)
All three corresponding sides are equal in length.
See Triangle Congruence (side side side).
- SAS (side angle side)
A pair of corresponding sides and the included angle are equal.
See Triangle Congruence (side angle side).
- ASA (angle side angle)
A pair of corresponding angles and the included side are equal.
See Triangle Congruence (angle side angle).
- AAS (angle angle side)
A pair of corresponding angles and a non-included side are equal.
See Triangle Congruence (angle angle side).
- HL (hypotenuse leg of a right triangle)
Two right triangles are congruent if the hypotenuse and one leg are equal.
See Triangle Congruence (hypotenuse leg).
AAA does not work.
If all the corresponding angles of a triangle are the same, the triangles will be the same shape, but not necessarily the same size.
For more on this see
Why AAA doesn't work.
They are called similar triangles (See Similar Triangles).
SSA does not work.
Given two sides and a non-included angle, it is possible to draw two different triangles that satisfy the the values.
It is therefore not sufficient to prove congruence.
See Why SSA doesn't work.
Another way to think about the above is to ask if it is possible to construct a unique triangle given what you know.
For example, If you were given the lengths of two sides and the included angle (SAS), there is only one possible triangle you could draw.
If you drew two of them, they would be the same shape and size - the definition of congruent.
For more on constructions, see Introduction to Constructions
Properties of Congruent Triangles
If two triangles are congruent, then each part of the triangle (side or angle) is congruent to the corresponding part in the other triangle.
This is the true value of the concept; once you have proved two triangles are congruent,
you can find the angles or sides of one of them from the other.
To remember this important idea, some find it helpful to use the acronym
which stands for "Corresponding Parts of Congruent Triangles are Congruent".
In addition to sides and angles, all other properties of the triangle are the same also, such as area, perimeter, location of centers, circles etc.
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