Question

$\huge \mathfrak \red{Question}$

Q.1: ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA. Prove that

(i) ΔABD ≅ ΔBAC
(ii) BD = AC
(iii) ∠ABD = ∠BAC.​

Answer 1

Given:

• ABCD is a quadrilateral
• AD = BC and ∠ DAB = ∠ CBA

To prove:

• Δ ABD ≅ Δ BAC
• BD = AC
• ∠ ABD = ∠ BAC

Proof:

Consider the Δ ABD and Δ BAC

Here,

→ AD = BC     [given]

→ ∠ DAB = ∠ CBA      [given]

→ AB = AB     [common]

So, using SAS congruency criterion

Δ ABD ≅ Δ BAC

Now,

By CPCT (corresponding parts of congruent triangles)

→ BD = AC  

also, By CPCT

→ ∠ ABD = ∠ BAC

PROVED.

_________________________________

SAS congruency rule (postulate)

• If two sides and the included angle in one triangle are congruent to two sides and the included angle in another triangle, then the two triangles are congruent. this is known as Side-angle-side (SAS) congruency rule.

CPCT theorem

• CPCT stands for Corresponding parts of Congruent triangles. CPCT theorem states that if two or more triangles which are congruent to each other are taken then the corresponding sides and the angles of the triangles are also congruent to each other.

Answer 2

Given:

ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA

‎

To Prove:

(i) ∆ABD ≅ ∆BAC

(ii) BD = AC

(iii) ∠ABD = ∠BAC

‎‎

Proof:

(i) In ∆ABD and ∆BAC,

AD = BC | Given

AB = BA | Common

∠DAB = ∠CBA | Given

∴ ∆ABD ≅ ∆BAC | SAS Cong. rule

‎

(ii) ∵ ∆ABD ≅ ∆BAC [proved in (i)]

∴ BD = AC | By CPCT

‎

(iii) ∵ ∆ABD ≅ ∆BAC [proved in (i)]

∴ ∠ABD = ∠BAC | By CPCT

$‎$