what is asa and sss and sas criterion and how  to detect them

The answer posted by isha.. is correct.

For more information on congruency rules, you can go through the content of chapter 7, lessons 2 to 6.

@isha..: Well done. Keep posting!!

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SSS congruence criterion
If three sides of a triangle are equal to the three corresponding sides of another triangle, 
then the triangles are congruent.
 
SAS congruence criterion
If two sides of a triangle and the angle included between them are equal to the 
corresponding two sides and included angle of another triangle, then the triangles are
congruent.
 
ASA congruence criterion
If two angles and included side of a triangle are equal to the two corresponding angles and 
the included side of another triangle, then the triangles are congruent.
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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 CriterionTwo 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 CriterionTwo 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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