**Congruent Triangles**

Two triangles are said to be **congruent triangles** if they are exactly the same by their **shapes** and **sizes**.

Two triangles of the **same shape** are meant three angles of one
triangle are equal to the corresponding three angles of another triangle. And,
two triangles of the **same size** are meant three sides of one triangle are equal
to the corresponding three sides of another triangle.

The symbol **≅** is used for congruent triangles.

For example:

In the given triangles ABC and PQR, AB = PQ, BC = QR, AC = PR.
Also, ∠A = ∠P, ∠B = ∠Q and ∠C = ∠R. So the triangles ABC
and PQR are congruent. It is written as ΔABC ≅ ΔPQR.

**Conditions for
Congruency**

The following conditions are necessary for the congruency of two triangles.

**Condition - 1:**** Side-Side-Side (SSS)
axiom**

When three sides of a triangle are equal to three corresponding sides of another triangle, they are congruent by Side-Side-Side condition. It is called SSS axiom.

In the given triangles, AB = PQ (S), BC = QR (S), AC = PR (S). Therefore, ΔABC ≅ ΔPQR by SSS axiom.

**Condition - 2:**** Side-Angle-Side
(SAS) axiom**

When Side-Angle-Side serially of a triangle are equal to the corresponding Side-Angle-Side serially of another triangle, then the triangles are congruent by Side-Angle-Side condition. It is called SAS axiom.

In the given triangles, AB = PQ (S), ∠B = ∠Q (A), BC = QR (S). Therefore, ΔABC ≅ ΔPQR by SAS axiom.

**Condition - 3:**** Angle-Side-Angle
(ASA) axiom**

When Angle-Side-Angle serially of a triangle are equal to the corresponding Angle-Side-Angle serially of another triangle, then the triangles are congruent by Angle-Side-Angle condition. It is called ASA axiom.

In the given triangles, ∠B = ∠Q (A), BC = QR (S), ∠C = ∠R (A). Therefore, ΔABC ≅ ΔPQR by ASA axiom.

**Condition - 4:**** Right angle-Hypotenuse-Side
(RHS) axiom**

In two right-angled triangles, if the hypotenuse and a side of a triangle are equal to the hypotenuse and a side of another triangle, then the triangles are congruent by Right angle-Hypotenuse-Side condition. It is called RHS axiom.

In the given triangles, ∠B = ∠Q = 90° (R), AC = PR (H), BC = QR (S). Therefore, ΔABC ≅ ΔPQR by RHS axiom.

**Condition - 5:**** Side-Angle-Angle (SAA)
axiom**

When Side-Angle-Angle serially of a triangle are equal to the corresponding Side-Angle-Angle serially of another triangle, then the triangles are congruent by Side-Angle-Angle condition. It is called SAA axiom.

In the given triangles, BC = QR (S), ∠C = ∠R (A), ∠A = ∠P (A). Therefore, ΔABC ≅ ΔPQR by SAA axiom.

**Worked Out Examples**

**Example 1:** In the given figure, state the condition of congruency and write
their corresponding sides and angles.

**Solution:** Here,

In ΔPQR and ΔABC

i.
∠B
= ∠Q
(A) -------> both 60°

ii.
BC = QR (S) -------> both 5cm

iii.
∠C
= ∠R
(A) -------> both 45°

∴ ΔABC ≅ ΔPQR
-------> by ASA axiom

∴ AB = PQ, AC = PR -------> corresponding sides of congruent triangles

∴ ∠A = ∠P -------> corresponding angles of congruent triangles

**Example 2:** In the given figure, BP = PC, AP⊥ BC, prove that ΔABP≅ ΔAPC, also find the value of a.

**Solution:** Here,

In ΔABP and ΔAPC

i.
BP = PC (S) -------> given

ii.
∠APB
= ∠APC
(A) -------> both right angles

iii.
AP = AP (S) -------> common side of both triangles

∴ ΔABP ≅ ΔAPC -------> by SAS axiom

∴ AB = AC -------> corresponding sides of congruent triangles

or, 2a + 3 = 5

or, 2a = 5 – 3

or, 2a = 2

or, a = 1
Ans.

**Example 3:** In the given figure, AP = DP, CP = BP, ∠PAC = 30° and ∠PDB = 40° then,

a.
Prove that ΔAPC≅ ΔDPB

b. Find the values of a, b and y.

**Solution:** Here,

In ΔAPC and ΔBPD

i.
AP = PD (S) -------> given

ii.
∠APC
= ∠DPB
(A) -------> vertically opposite angles

iii.
PC = PB (S) -------> given

∴ ΔAPC ≅ ΔDPB --------> by SAS axiom

∴ a = 40° ------->
corresponding angles of congruent triangles

∴ b = 30° ------->
corresponding angles of congruent triangles

∴ AC = DB -------> corresponding sides of
congruent triangles

or, y + 3 = 2y

or, 3 = 2y – y

or, y = 3
Ans.

If you have any questions or problems regarding the **Congruent Triangles**, you can ask here, in the comment section below.

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