TRANSITIVE PROPERTY OF CONGRUENCE

When two shapes or figures have the same shape and size, we use the term congruence and they shapes or figures are said to be congruent (≅). 

Example :

In the diagram given below, Triangle MQN is congruent to triangle ABC. Prove that triangle PQR is congruent to triangle ABC. 

Solution :

Given : Triangle MQN is congruent to triangle ABC.

In the above diagram, we do not have any details about the triangle ABC.

So, if we prove triangle PQR is congruent to MQN, then we can prove triangle PQR is congruent to triangle ABC using transitive property of congruent triangles.

Proving triangle PQR is congruent to triangle MQN :

From the above diagram, we are given that all three pairs of corresponding sides of triangle PQR and MQN are congruent. 

That is

RP ≅ MN, PQ ≅ NQ and QR ≅ QM

Because ∠P and ∠N have the same measure, ∠P ≅ ∠N.

By the Vertical Angles Theorem, we know that 

ΔPQR ≅ ΔMQN

By the Third Angles Theorem, if two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

So, we have 

∠R ≅ ∠M

So, all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. By the definition of congruent angles, 

ΔPQR ≅ ΔMQN

Using Transitive Property of Congruent Triangles :

By Transitive property of congruent triangles, 

if ΔPQR ≅ ΔMQN and ΔMQN ≅ ΔABC, then 

ΔPQR ≅ ΔABC

Thus, triangle PQR is congruent to triangle ABC. 

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