Question

ABCD is a quadrilateral in which AD=BC and <DAB=<CBA (see Fig).Prove that
(i)∆ABD=~∆BAC
(ii)BD=AC
(iii)<ABD=<BAC​

Answer 1

Correct question :

ABCD is a quadrilateral in which AD = BC and

∠ DAB = ∠ CBA Prove that

(i) ∆ ABD ≅ ∆ BAC

(ii) BD = AC

(iii) ∠ ABD = ∠ BAC.

Answer :

GiveN :

• AD = BC
• ∠ DAB = ∠ CBA

To prove :

• ∆ ABD ≅ ∆ BAC
• BD = AC
• ∠ ABD = ∠ BAC.

Solution :

❦__________________________❦

$\large{\bf{\boxed{\green{1→}}}}$

In both ∆ABD and ∆ BAC, We have :

AD = BC ---------( given )

∠DAB = ∠CBA ---------( given )

AB = AB ---------( Common in both ∆ )

Therefore,

In both triangles, Two sides and one angle of one triangle is equal to two sides and one angle of another triangle

Hence,

$\small{\boxed{\bf{\pink{∆ ABD ≅ ∆ BAC\: ---(proved\: by\: SAS \: congruency\: rule*}}}}$

❦__________________________❦

$\large{\bf{\boxed{\green{2→}}}}$

In ∆ ABD and ∆ BAC :

• ∆ ABD ≅ ∆ BAC ---------(proved)

Therefore ,

BD = AC -------( CPCT* )

$\large{\boxed{\bf{\pink{BD=AC -----(proved)}}}}$

❦__________________________❦

$\large{\bf{\boxed{\green{3→}}}}$

In ∆ ABD and ∆ BAC :

• ∆ ABD ≅ ∆ BAC ---------(proved)

Therefore,

∠ ABD = ∠ BAC -----(CPCT)

$\large{\boxed{\bf{\pink{∠ABD =∠BAC----(proved)}}}}$

❦__________________________❦

❦__________________________❦

Additional information:

* SAS congruency = If any two sides and one angle of one triangle is equal to any two sides and one angle of another triangle, then both triangle are said to be congruence by ' SAS rule '

* CPCT = CPCT stands for Corresponding part of congruent triangle.

If two triangles are congruent ,

• Their corresponding sides are equal
• Their corresponding angles are equal