Congruent  Triangles

If two triangles are equal in all respects, they are said to be
Congruent triangles .
Thus two congruent-triangles have the same shape and same size.


Let ΔABC and ΔPQR be two triangles. Then we can superimpose ΔABC on ΔPQR, so as to cover exactly.

Due to this superimposition :
Vertex A falls on Vertex P
Vertex B falls on Vertex Q
Vertex C falls on Vertex R
AB = PQ ∠A = ∠P
BC =QR ∠B = ∠Q
AC = PR ∠C = ∠R
Hence triangles ABC and PQR are congruent to each other.

Note : 1) Congruent-triangles are similar but the similar triangles are not always congruent.
2) The symbol
reads " is congruent to ".

If two triangles are congruent then there is
one to one correspondence (↔) between the two triangles.

ΔABC ↔ ΔPQR then

∠A ≅ ∠P , ∠B ≅ ∠Q and ∠C ≅ ∠R

∠AB ≅ ∠PQ , ∠BC ≅ ∠QR and ∠AC ≅ ∠PR .

Note : If two triangles are congruent then their corresponding parts are congruent.

C orresponding P arts of C ongruent T riangles are C ongruent
⇒ C. P .C. T. C

Congruence Relation

1) Every triangle is congruent to itself. ΔABC ≅ ΔABC.
2) If ΔABC ≅ ΔPQR then ΔPQR = ΔABC.
3) If ΔABC ≅ ΔPQR and ΔABC ≅ ΔDEF then ΔPQR ≅ ΔDEF.

Examples :

In the following pairs of triangles, find out whether the triangles in each pair are congruent or not.
1) ΔABC : AB = 3 , BC = 4 and ∠B = 90
0
ΔDEF : DE = 3 , DF = 4 and ∠E = 90
0 .

Solution :
Here , ΔABC
not ≅ Δ DEF because there is no one to one correspondence between BC and DF.

2) Δ ABC : AB = 3 , AC = 5 and BC = 6
Δ PQR : PQ = 3 , PR = 5 and QR = 6

Solution :
Here ΔABC
ΔPQR because there is one to one correspondence between all the sides.


Triangles

Introduction to Triangles
Types of Triangles on the basis of Sides
Types of Triangles on the basis of Angles
Angle Sum Property of Triangles
Exterior and Interior angles of Triangle
Triangle Inequality Property
Congruent Triangles
Postulates of Congruent Triangle
Inequality in Triangle

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