PROPERTIES OF CONGRUENT TRIANGLES

In this section, you will learn the congruent triangles properties which will be useful to verify whether two triangles are congruent or not.

Reflexive Property of Congruent Triangles

Every triangle is congruent to itself.

In the diagram above, triangle ABC is congruent to it self.

Symmetry Property of Congruent Triangles

In the diagram above,

if ΔABC  ≅  ΔDEF, then 

ΔDEF  ≅  ΔABC

Transitive Property of Congruent Triangles

In the diagram above, 

if ΔABC  ≅  ΔDEF and ΔDEF  ≅  ΔJKL, then 

ΔABC  ≅  ΔJKL

Solved Problems

Problem 1 :

Prove the Reflexive Property of Congruent Triangles.

Solution :

If two triangle are considered to be congruent, they have to meet the following two conditions.

1. They must have exactly the same three sides.

2. They must have exactly the same three angles.

Every triangle and itself will meet the above two conditions. 

So, every triangle is congruent to itself. 

Problem 2 :

In the diagram given below, triangle ABD is congruent to triangle BCD. Is triangle BCD congruent to triangle ABC ? Explain your reasoning. 

Solution :

Yes, triangle BCD is congruent to triangle ABC. 

By Symmetry Property of Congruent Triangles, 

if ΔABD ≅ ΔBCD, then 

ΔBCD ≅ ΔABD. 

Thus, triangle BCD is congruent to triangle ABC

Problem 3 :

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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