Question

From the given information, specify the type of quadrilateral ABCD in each
case:
i) AB║CD, AD║BC, ∠D = ∠A
ii) AB║CD, AD = CB
iii) AB║CD, BC = CD
iv) AB║CD, AD║BC, AB = CD, BC =AD, AD ┴ AB

Answer 1

$\huge\fbox \red{A}\fbox \green{N}\fbox \purple{S}\fbox \orange{W}\fbox \red{E}\fbox \blue{R}$

Step-by-step explanation:

In quadrilateral ABCD, we have AB=CD and AD=BC

⇒ Now, join segment AC.

⇒ In △ABC and △ACD, we have

⇒ AB = CD [Given]

⇒ AD = BC [Given]

⇒ AC = AC [Common side]

So, by SSS criteria,

⇒ △ABC≅△ACD

∴ ∠DAC = ∠BCA [By CPCT]

∴ CD∥AB [∵ Alternate angles are equal] ---- ( 1 )

⇒ ∠DCA=∠CAB [By CPCT]

∴ AD∥BC [∵ Alternate angles are equal] --- ( 2 )

From ( 1 ) and ( 2 ) we get that opposite sides of quadrilateral are parallel.

∴ ABCD is a parallelogram..

Answer 2

$\huge\fbox \red{A}\fbox \green{N}\fbox \purple{S}\fbox \orange{W}\fbox \red{E}\fbox \blue{R}$

Step-by-step explanation:

In quadrilateral ABCD, we have AB=CD and AD=BC

⇒ Now, join segment AC.

⇒ In △ABC and △ACD, we have

⇒ AB = CD [Given]

⇒ AD = BC [Given]

⇒ AC = AC [Common side]

So, by SSS criteria,

⇒ △ABC≅△ACD

∴ ∠DAC = ∠BCA [By CPCT]

∴ CD∥AB [∵ Alternate angles are equal] ---- ( 1 )

⇒ ∠DCA=∠CAB [By CPCT]

∴ AD∥BC [∵ Alternate angles are equal] --- ( 2 )

From ( 1 ) and ( 2 ) we get that opposite sides of quadrilateral are parallel.

∴ ABCD is a parallelogram..