**Properties of Congruent Triangles Worksheet :**

Worksheet given in this section is much useful to the students who would like to practice problems on properties of congruent triangles.

To know about "Properties of Congruent Triangles" in detail,

**Problem 1 :**

Prove the Reflexive Property of Congruent Triangles.

**Problem 2 :**

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

**Problem 3 :**

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

**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.

After having gone through the stuff given above, we hope that the students would have understood, "Properties of congruent triangles worksheet".

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