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see asa means side angle side

in this if 2angles and one side is equal of two triangle we can say that the triangle is congurent b asa rule

sss means side side side

if all the3 sides are equal we can say both the triangle is congurent by sss rule

sas means side angle side

in this if one angles and 2 side is equal of two triangle we can say that the triangle is congurent b sas rule

nitish Mittal

VII-'A'

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__ASA Criterion__: Two

**triangles**are

**congruent**if

**two**

**angles**and the included side of one

**triangle**are equal to the

**corresponding**

**two**

**angles**and the

**included**

**side**of the other

**triangle**.

__SAS Criterion__:

**Two**

**triangles**are

**congruent**if two sides and the

**included**

**angle**of one triangle are equal to the

**corresponding**

**two**

**sides**and the

**included**

**angle**of the other

**triangle**.

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**SSS**

We use several criteria to prove two triangles to be congruent. The most common criterion is the Side-Side-Side (SSS) congruency criterion and is applied when we know all the sides of the two triangles.

**Example 1:**

**Is the following pair of triangles congruent?**

**Solution:**

In ΔABC and ΔPQR,

Therefore, ΔABC and ΔPQR are not congruent triangles.

**Example 2:**

**In the following figure, and D is the mid-point of BC.**

**(1) Prove that ΔACD ≅ΔABD.**

**(2) Show that ∠CAD = ∠BAD.**

**Solution:**

**(1) **In ΔACD and ΔABD,

(Given)

(Given D is the mid-point of BC)

(Common side)

Therefore, ΔACD ≅ ΔABD (By SSS congruence criterion)

**(2)** ΔACD ≅ ΔABD {By SSS congruence criterion}

Therefore, ∠CAD = ∠BAD {Corresponding angles of congruent triangles}

**SAS**

We know that if the two figures are of same shape and size, then the two figures are congruent to each other. If all the three sides of both the triangles are given, then we can easily determine whether the triangles are congruent or not.

Consider the two triangles, ΔDEF and ΔXYZ, as shown in the following figure.

In these triangles,

And, ∠DEF = ∠XYZ

Thus, under the correspondence DEF ↔ XYZ, using SAS congruency criterion, we obtain

ΔDEF ≅ ΔXYZ

Now, consider the following triangles where in ΔABC, = 8.4 cm, = 6.8 cm, and ∠B = 65°. In ΔXYZ, = 6.8 cm, =8.4 cm, and ∠X = 65°. Can we say “the two triangles are congruent”?

In ΔABC and ΔXYZ,

∠B = ∠X = 65°

Here, two sides and their included angle of ΔABC are equal to their two corresponding sides and their included angle of ΔXYZ under the correspondence ABC ↔ ZXY.

Therefore, according to SAS congruence criterion,

ΔABC ≅ ΔZXY

Let us now solve some more examples.

**Example1:**

**Which of the following pairs of triangles are congruent?**

**(i)**

**(ii)**

**Solution:**

**(i)** In ΔABC and ΔDEF,

The included angle between and is ∠B, and the included angle between and is ∠E.

It is given that, ∠B = 47° and ∠E = 50°

⇒ ∠B ≠ ∠E

Thus, the included angle between the two sides is not same.

Therefore, ΔABC and ΔDEF are not congruent.

**(ii)** In ΔABC and ΔMNL,

The angle included between and is ∠C, and the angle included betweenand is ∠L.

It is given that, ∠C = ∠L = 60°

Therefore, according to SAS congruence criterion, we can say that ΔABC and ΔLMN are congruent.

Thus, we can write ΔABC ≅ ΔMNL.

**Example 2: **In the figure shown here, and ∠QPR = ∠PQS.

Prove that ΔPQR ≅ ΔQPS.

Show that and ∠QPS = ∠PQR

**Solution:**

In ΔPQR and ΔQPS,

(given)

∠QPR = ∠PQS (given)

PQ = PQ (common side)

Therefore, ΔPQR ≅ ΔQPS (by SAS congruence criterion)

Thus, (corresponding sides of congruent triangles)

∠QPS = ∠PQR (corresponding angles of congruent triangles)

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