Question

In the given figure the congruency rule used in proving angle ACB congrats to angle ADB​

Answer 1

Given:-

• Two triangles are given to us.
• The triangles are ∆ACB and ∆ADB .

To Prove:

• $\sf{\triangle ACB \cong \triangle ADB }$

Proof:

Here in ∆ACB and ∆ADB ,

• AB = AB ( common )
• AC = AD ( given)
• $\sf{\angle}$CAB = $\sf{\angle}$DAB ( given)

Hence by SAS congruence condition we can say that $\sf{\triangle ACB \cong \triangle ADB }$.

More to Know:-

There are 5 congruence conditions :-

• SAS ( Side angle Side)
• ASA ( Angle Side Angle )
• AAS ( Angle Angle Side)
• SSS( Side Side Side)
• RHS ( Right angle Hypontenuse Side)

i) SAS :

This condition is an axiom using which we Prove other congruence conditions .

In this triangles are said to be congruent if two sides and the angle included between the two sides are equal.

ii) ASA :

In this two triangles are said to be congruent if two pair of angles are equal and the side and inclosed between the two angle is equal.

iii) AAS :

In this two two triangles are said to be congruent if any two pair of angles and any corresponding side is equal.

iv) SSS:

In this congruence condition if all the three sides of the triangles are equal then the triangles are said to be congruent .

v) RHS:-

If two right-angled triangles have their hypotenuses equal in length, and a pair of shorter sides are equal in length, then the triangles are congruent.

Answer 2

Given :–

• AC = AD
• ∠CAB = ∠DAB

To Prove :–

• ∆ACB ≅ ∆ADB

Proof :–

In ∆ACB and ∆ADB,

• AC = AD [Given]

• ∠CAB = ∠DAB [Given]

• AB = AB [Common Side]

By SAS (Side Angle Side) congruence rule.

∆ACB ≅ ∆ADB

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Additional Information :–

Some simple congruence theorems :-

1) ASA = Angle Side Angle.

In this theorem, two angles and 1 side of two triangles are equal.

Lets, take one example,

We have 2 triangle ∆ABC and ∆PQR.

So,

In ∆ABC and ∆PQR,

• ∠BAC = ∠QPR

• AC = PR

• ∠ACB = ∠PRQ

Then, ∆ABC ≅ ∆PQR.

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2) SSS = Side Side Side.

In this theorem, all the three sides of two triangles are equal.

Example :-

We have 2 triangles ∆ABC and ∆PQR.

So,

In ∆ABC and ∆PQR,

• AB = PQ

• BC = QR

• AC = PR

Then, ∆ABC ≅ ∆PQR.

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3) AAS = Angle Angle Side.

In this theorem, two angles and one side of two triangles are equal.

Example :-

We have 2 triangles ∆ABC and ∆PQR.

So,

In ∆ABC and ∆PQR,

• ∠BAC = ∠QPR

• ∠ACB = ∠PRQ

• BC = QR

Then, ∆ABC ≅ ∆PQR

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4) RHS = Right-angle Hypotenuse Side.

In this theorem, We need two right angled triangle, (right angle) ∠90° are equal of two triangles, one hypotenuse is equal to the other triangle and one side of two triangles are equal.

Example :-

In ∆ABC = ∆PQR

• ∠B = ∠Q [both 90° (Right angle)]

• AC = PR [Hypotenuse]

• BC = QR [Side]