Congruency of triangles

Difference between congruency and similarity

If two or more figures have exactly the same shape and size, then they are known to be congruent.

The above figures have exactly the same shape and size; the second one is just rotated 90° clockwise, and the third one is horizontally flipped. So they are congruent.

Two figures are similar, if they have exactly the same shape but not necessarily of the same size.

The above figures look the same, but they difffer in size. The second one is scaled-up, and the third one is scaled-down image of the first one. So the figures are similar, but not congruent.

Corresponding Parts of Congruent Triangles are equal

Two or more triangles are congruent if their
three pairs of corresponding angles are congruent,
and three pairs of corresponding sides are congruent.

In other words, we say that if two or more triangles are congruent,
then their corresponding parts are equal.
The term CPCT stands for "Corresponding Parts of Congruent Triangles".

In the above, all the four triangles are congruent, because
\(AB=DE=QR=ZX \)  (corresponding sides)
\(BC=EF=RP=XY \)      -do-
\(CA=FD=PQ=YZ \)      -do-
\(∠A=∠D=∠Q=∠Z \)  (corresponding angles)
\(∠B=∠E=∠R=∠X \)      -do-
\(∠C=∠F=∠P=∠Y \)      -do-

Five conditions of congruency in triangles

For two triangles to be congruent, one of the following five conditions must be met:

• ● SSS (Side-Side-Side)
• ● SAS (Side-Angle-Side)
• ● ASA (Angle-Side-Angle)
• ● AAS (Angle-Angle-Side)
• ● RHS (Right angle-Hypotenuse-Side)

Side-Side-Side (SSS)

The term SSS means three sides of a triangle.
If three sides of one triangle are equal to the corrresponding three sides of another triangle, then both the triangles are congruent.

In the above it is given that \(AB=DE,\,BC=EF \) and \(CA=FD\). So, both the triangles are congruent by SSS criteria, and \(\angle A=\angle D,\,\angle B=\angle E,\) and \(\angle C=\angle F\).

Note: No matter in what orientation you draw them, they are congruent because corresponding sides are equal.

Side-Angle-Side (SAS)

By SAS we mean two sides and the included angle of a triangle.
The term included angle refers to the angle created by the two given sides.
As per SAS criteria, two triangles are congruent if the two sides and the included angle of one triangle are equal to the corresponding sides and the included angle of the other triangle.

In the above it is given that two sides and the included angle of one triangle are equal to the two sides and the included angle of another triangle, i.e., \(CA=FD,\,AB=DE \) and \(∠A=∠D\). So, both the triangles are congruent by SAS criteria, and \(\angle B=\angle E,\,\angle C=\angle F,\) and \(BC=EF\).

Angle-Side-Angle (ASA)

By ASA we mean two angles and the included side of a triangle.
The term included side refers to the side surrounded by the two given angles.
As per ASA criteria, two triangles are congruent if the two angles and the included side of one triangle are equal to the corresponding angles and the included side of the other triangle.

In the above it is given that two angles and the included side of one triangle are equal to the two angles and the included side of another triangle, i.e., \(∠A=∠D, ∠B=∠E, \) and \(AB = DE\). So, both the triangles are congruent by ASA criteria, and \(\angle C=\angle F,\,BC=EF\) and \(CA=FD\).

Angle-Angle-Side (AAS)

The term AAS means two angles and an non-included side of a triangle.
As per AAS criteria, two triangles are congruent if two angles and a non-included side of one triangle are equal to the corresponding angles and the non-included side of the other triangle.

In the above it is given that two angles and a non-included side (BC) of one triangle are equal to the two angles and the non-included side (EF) of another triangle, i.e., \(∠A=∠D, ∠B=∠E, \) and \(BC = EF\). So, both the triangles are congruent by AAS criteria, and \(\angle C=\angle F,\,AB=DE\) and \(CA=FD\).

Right angle-Hypotenuse-Side (RHS)

The term RHS means the hypotenuse and any one side of a right-angled triangle.
As per RHS criteria, two right-angled triangles are congruent if the hypotenuse and any one side of one triangle are equal to the hypotenuse and the corresponding side of the other triangle.

In the above it is given that both the hypotenuses are equal (\(BC\,=\,DE\)), and also given \(CA\,=\,EF\).
So according to RHS criteria, \(\triangle ABC \cong \triangle DEF\), i.e., \(\angle C=\angle E,\, \angle B=\angle D,\, AB=FD\).

Congruency of triangles: is the SSA rule invalid?

SSA criteria of triangle congruency is not valid, and sometimes called ambiguous, because with given two sides and a non-included angle sometimes two different triangles can be drawn. Kindly check this page for more details.

You may also refer to the page Triangle Calculator where you can get any triangle drawn by providing suitable input data like three sides(SSS), two sides and the angle inbetween (SAS) etc.

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