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Criteria for Congruence of Triangles - SAS Congruence Criterion

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definition

SAS Congruence criterion: If under a correspondence, two sides and the angle included between them of a triangle are equal to two corresponding sides and the angle included between them of another triangle, then the triangles are congruent.

notes

SAS Congruence criterion:

If under a correspondence, two sides and the angle included between them of a triangle are equal to two corresponding sides and the angle included between them of another triangle, then the triangles are congruent.

Which congruence criterion do you use in the following?

Given:
ZX = RP
RQ = ZY
∠PRQ = ∠XZY
So, ∆PQR ≅ ∆XYZ

Solution: SAS Congruence criterion, as two sides and the angle included between these sides of ΔPQR are equal to two sides and the angle included between these sides of ΔXYZ.

Example

Given below are measurements of some parts of two triangles. Examine whether the two triangles are congruent or not, by using the SAS congruence rule. If the triangles are congruent, write them in symbolic form.

∆ABC, AB = 7 cm, BC = 5 cm, ∠B = 50°.

∆DEF, DE = 5 cm, EF = 7 cm, ∠E = 50°.

Here,
AB = EF ( = 7 cm),
BC = DE ( = 5 cm) and
included ∠B = included ∠E ( = 50°).

Also, A ↔ F B ↔ E and C ↔ D.

Therefore, ∆ABC ≅ ∆FED................(By SAS congruence rule).

Example

Given below are measurements of some parts of two triangles. Examine whether the two triangles are congruent or not, by using the SAS congruence rule. If the triangles are congruent, write them in symbolic form.

∆ABC, AB = 4.5 cm, AC = 4 cm, ∠A = 60°.

∆DEF, DE = 4 cm, FD = 4.5 cm, ∠D = 55°.

Here, AB = FD and AC = DE.

But included ∠A ≠ included ∠D.

So, we cannot say that the triangles are congruent.

Example

In Fig,

AB = AC and AD is the bisector of ∠BAC.
(i) State three pairs of equal parts in triangles ADB and ADC.
(ii) Is ∆ADB ≅ ∆ADC? Give reasons.
(iii) Is ∠B = ∠C? Give reasons.

(i) The three pairs of equal parts are as follows:
AB = AC...................(Given)
∠BAD = ∠CAD......(AD bisects ∠BAC) and
AD = AD..................(common)

(ii) Yes, ∆ADB ≅ ∆ADC.....(By SAS congruence rule)

(iii) ∠B = ∠C.........................(Corresponding parts of congruent triangles)

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