NAME: __________________
DATE: _______
Chapter 12:
Side-side-side (SSS) CONGRUENCE
AND
Side-angle-side (SAS) CONGRUENCE
Congruence of Triangles
Congruence of triangles: Two triangles are said to be congruent if all three corresponding sides
are equal and all the three corresponding angles are equal in measure. These triangles can be
slides, rotated, flipped and turned to be looked identical. If repositioned, they coincide with each
other. The symbol of congruence is’ ≅’.
The meaning of congruence in Math is when two figures are similar to each other based on their
shape and size. There are basically four congruence rules that proves if two triangles are
congruent. But it is necessary to find all six dimensions. Hence, the congruence of triangles can
be evaluated by knowing only three values out of six. The corresponding sides and angles of
congruent triangles are equal. Also, learn about Congruent Figures here.
Congruence is the term used to define an object and its mirror image. Two objects or shapes are
said to be congruent if they superimpose on each other. Their shape and dimensions are the
same. In the case of geometric figures, line segments with the same length are congruent and
angle with the same measure are congruent.
Conditions for Congruence of Triangles:

SSS (Side-Side-Side)

SAS (Side-Angle-Side)

ASA (Angle-Side-Angle)

AAS (Angle-Angle-Side)

RHS (Right angle-Hypotenuse-Side)
CPCT is the term, we come across when we learn about the congruent triangle. Let’s see the
condition for triangles to be congruent with proof.
Congruent meaning in Maths
NAME: __________________
DATE: _______
The meaning of congruent in Maths is addressed to those figures and shapes that can be
repositioned or flipped to coincide with the other shapes. These shapes can be reflected to
coincide with similar shapes.
Two shapes are congruent if they have the same shape and size. We can also say if two shapes
are congruent, then the mirror image of one shape is the same as the other.
Congruent Triangles
A closed polygon made of three line segments forming three angles is known as a Triangle.
Two triangles are said to be congruent if their sides have the same length and angles have same
measure. Thus, two triangles can be superimposed side to side and angle to angle.
In the above figure, Δ ABC and Δ PQR are congruent triangles. This means,
Vertices: A and P, B and Q, and C and R are the same.
Sides: AB=PQ, QR= BC and AC=PR;
Angles: ∠A = ∠P, ∠B = ∠Q, and ∠C = ∠R.
Congruent triangles are triangles having corresponding sides and angles to be equal. Congruence
is denoted by the symbol “≅”. From the above example, we can write ABC ≅ PQR. They have
the same area and the same perimeter.
CPCT Rules in Maths
The full form of CPCT is Corresponding parts of Congruent triangles. After proving triangles
congruent, the remaining dimension can be predicted without actually measuring the sides and
angles of a triangle. Different rules of congruency are as follows.




SSS (Side-Side-Side)
SAS (Side-Angle-Side)
ASA (Angle-Side-Angle)
AAS (Angle-Angle-Side)
NAME: __________________
DATE: _______

RHS (Right angle-Hypotenuse-Side)
Let us learn them all in detail.
SSS (Side-Side-Side)
If all the three sides of one triangle are equivalent to the corresponding three sides of the second
triangle, then the two triangles are said to be congruent by SSS rule.
In the above-given figure, AB= PQ, BC = QR and AC=PR, hence Δ ABC ≅ Δ PQR.
SAS (Side-Angle-Side)
If any two sides and the angle included between the sides of one triangle are equivalent to the
corresponding two sides and the angle between the sides of the second triangle, then the two
triangles are said to be congruent by SAS rule
In above given figure, sides AB= PQ, AC=PR and angle between AC and AB equal to angle
between PR and PQ i.e. ∠A = ∠P. Hence, Δ ABC ≅ Δ PQR.
ASA (Angle-Side- Angle)
If any two angles and the side included between the angles of one triangle are equivalent to the
corresponding two angles and side included between the angles of the second triangle, then the
two triangles are said to be congruent by ASA rule.
In above given figure, ∠ B = ∠ Q, ∠ C = ∠ R and sides between ∠B and ∠C , ∠Q and ∠ R are
equal to each other i.e. BC= QR. Hence, Δ ABC ≅ Δ PQR.
NAME: __________________
DATE: _______
AAS (Angle-Angle-Side) [Application of ASA]
AAS stands for Angle-Angle-Side. When two angles and a non-included side of a triangle are
equal to the corresponding angles and sides of another triangle, then the triangles are said to be
congruent.
AAS congruence can be proved in easy steps.
Suppose we have two triangles ABC and DEF, where,
∠B = ∠E [Corresponding angles]
∠C = ∠F [Corresponding angles] And
AC = DF [Adjacent sides]
By angle sum property of triangle, we know that;
∠A + ∠B + ∠C = 180 ………(1)
∠D + ∠E + ∠F = 180 ……….(2)
From equation 1 and 2 we can say;
∠A + ∠B + ∠C = ∠D + ∠E + ∠F
∠A + ∠E + ∠F = ∠D + ∠E + ∠F [Since, ∠B = ∠E and ∠C = ∠F] ∠A = ∠D
Hence, in triangle ABC and DEF,
∠A = ∠D
AC = DF
∠C = ∠F
Hence, by ASA congruency,
Δ ABC ≅ Δ DEF
RHS (Right angle- Hypotenuse-Side)
If the hypotenuse and a side of a right- angled triangle is equivalent to the hypotenuse and a side
of the second right- angled triangle, then the two right triangles are said to be congruent by RHS
rule.
NAME: __________________
In above figure, hypotenuse XZ = RT and side YZ=ST, hence ∆ XYZ ≅ ∆ RST.
DATE: _______
NAME: __________________
DATE: _______
Solved Example
Let’s Work Out:
Example 1: In the following figure, AB = BC and
AD = CD. Show that BD bisects AC at right angles.
Solution: We are required to prove ∠BEA = ∠BEC
= 90° and AE = EC.Consider ∆ABD and ∆CBD,AB = BC
(Given)AD = CD
(Given)BD = BD
(Common)Therefore, ∆ABD ≅ ∆CBD
(By SSS congruency)∠ABD = ∠CBD
Now, consider ∆ABE and ∆CBE,
AB = BC
(Given)
∠ABD = ∠CBD
(Proved above)
BE = BE
Therefore, ∆ABE≅ ∆CBE
∠BEA = ∠BEC
And ∠BEA +∠BEC = 180°
2∠BEA = 180°
(Common)
(By SAS congruency)
(CPCTC)
(Linear pair)
(∠BEA = ∠BEC)
∠BEA = 180°/2 = 90° = ∠BEC
AE = EC
(CPCTC)
(CPCTC)
Hence, BD is a perpendicular bisector of AC.
NAME: __________________
Example 2: In a Δ ABC, if AB = AC and ∠ B = 70°, find ∠ A.
Solution: Given: In a Δ ABC, AB = AC and ∠B = 70°
∠ B = ∠ C [Angles opposite to equal sides of a triangle are equal]
Therefore, ∠ B = ∠ C = 70°
Sum of angles in a triangle = 180°
∠ A + ∠ B + ∠ C = 180°
∠ A + 70° + 70° = 180°
∠ A = 180° – 140°
∠ A = 40°
DATE: _______