Congruency and Similarity: The Difference
 Nov 29, 2017
 LearnPick
Most students develop this annoying tendency of mixing up congruency with similarity in their middle school geometrical problems.
If you also face the same problem with your math homework, you have come to the right place at the right time. In here, we have discussed the differences between similarity and congruency (with reference only to triangles) in meticulous details for your reference. Have a look.
CONGRUENT TRIANGLES
Mathematical definition
Two triangles are said to be congruent only if their corresponding angles and sides are congruent (or equal) to one another.
Here’s an example for your reference:
Source Wiki
According to the diagram above, both of the triangles are equal in shape and size. Hence, we can easily conclude that both of the triangles are congruent to one another. Mathematically, we represent congruent triangles as:
[Symbol Triangle] ABC ≅ [Symbol Triangle] DEF
How to understand congruency the easy way

Imagine you have two triangles made of cardboard.

You are placing one triangle on top of the other.

If it is a perfect fit, both of the triangles are congruent to one another.
Properties of congruency

If 2 triangles are congruent to one another, each of their corresponding angles and side must be congruent to one another.

Corresponding parts (sides or angles) of congruent triangles must be congruent to one another.
How to determine congruency of triangles
Triangular congruency can be proved by the following conditions:

SSS (Side, Side, Side): According to this condition, the 3 corresponding sides of a triangle must be congruent to one another.

ASA (Angle, Side, Angle): 2 corresponding angles must be congruent. The corresponding side in between the two angles must also be congruent.

SAS (Side, Angle, Side): 2 corresponding sides must be congruent. The corresponding angle in between must also be congruent.

AAS (Angle, Angle, Side): A pair of corresponding angles and a noninclusive side of a triangle must be congruent to the same things as that of the other.

HL (Hypotenuse and leg of a rightangled triangle)[Applicable only when the given angles are rightangled triangles].
SIMILAR TRIANGLES
Mathematical definition
2 triangles are considered similar to one another if both of them are identical in shapes, but are of different sizes.
Note: Even if the triangles are rotated in their respective position, the property of similarity must be retained at all possible cost.
For example:
Source Wiki
According to the image above, we have 2 triangles having identical shapes but nonidentical sizes. Thus, we can conclude that the given two triangles are similar to one another.
Mathematically, similar triangles are represented as:
[Symbol Triangle] ABC ~ [Symbol Triangle] GEF.
Properties of similarity

All corresponding angles of the two similar triangles must be equal. According to the diagram of the two similar triangles depicted above, we can say:
∠ABC = ∠EGF,
∠BAC = ∠GEF.
∠ACB = ∠EFG.

All corresponding sides of similar triangles should be in identical proportions. According to the diagram of the two similar triangles depicted above, we can say:
How to determine similarities between two triangles
According to the “similar triangles” image depicted above, each triangle exhibits six forms of basic measurement values:

Three sides, and

Three angles.
You don’t need to know the value of each of the three sides and angles of the triangles to determine the condition of similarity between the two (if any). Just a fulfillment of a few conditions would suffice. These conditions are:

AAA (Angle, Angle, Angle): Three corresponding angles of the two triangles must be similar to one another.

SSS (Side, Side, Side): Three corresponding sides of the two triangles must be in the same proportion.

SAS (Side, Angle, Side): 2 pairs of sides should always be in the same proportion. The corresponding angle included between the 2 sides must also be equal to one another.
If you are able to fulfill at least one condition among the three highlighted above, you can prove that the two triangles are indeed similar to one another.
So that sums up our discussion for now. Hope you had an enjoyable and an enlightening read. Cheerio!
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