Question

ii) If the diagonals of a parallelogram are equal, the parallelogram is a rectangle-
(Prove that A DAB = A CBA, and ZA+ ZB = 2 rt. Zs)​

Answer 1

□ ABCD is a parallelogram

consider Δ ACD and Δ ABD

AC = BD .... (given)

AB = DC .... (opposite sides of parallelogram)

AD = AD .... (common side)

∴Δ ACD ≅Δ ABD (sss test of congruence)

∠ BAD = ∠ CDA .... (cpct)

∠BAD+∠CDA=180

∘

. [Adjacent angles of parallelogram are supplementary]

so ∠ BAD and ∠ CDA are right angles as they are congruent and supplementary.

Therefor, □ ABCD is a rectangle since a

parallelogram with one right interior angle is a rectangle.

Answer 2

Answer:

Ref.Image

Ref.Image□ ABCD is a parallelogram

Ref.Image□ ABCD is a parallelogramconsider Δ ACD and Δ ABD

Ref.Image□ ABCD is a parallelogramconsider Δ ACD and Δ ABDAC = BD .... (given)

Ref.Image□ ABCD is a parallelogramconsider Δ ACD and Δ ABDAC = BD .... (given)AB = DC .... (opposite sides of parallelogram)

Ref.Image□ ABCD is a parallelogramconsider Δ ACD and Δ ABDAC = BD .... (given)AB = DC .... (opposite sides of parallelogram)AD = AD .... (common side)

Ref.Image□ ABCD is a parallelogramconsider Δ ACD and Δ ABDAC = BD .... (given)AB = DC .... (opposite sides of parallelogram)AD = AD .... (common side)∴Δ ACD ≅Δ ABD (sss test of congruence)

Ref.Image□ ABCD is a parallelogramconsider Δ ACD and Δ ABDAC = BD .... (given)AB = DC .... (opposite sides of parallelogram)AD = AD .... (common side)∴Δ ACD ≅Δ ABD (sss test of congruence)∠ BAD = ∠ CDA .... (cpct)

Ref.Image□ ABCD is a parallelogramconsider Δ ACD and Δ ABDAC = BD .... (given)AB = DC .... (opposite sides of parallelogram)AD = AD .... (common side)∴Δ ACD ≅Δ ABD (sss test of congruence)∠ BAD = ∠ CDA .... (cpct)∠BAD+∠CDA=180∘. [Adjacent angles of parallelogram are supplementary]

so ∠ BAD and ∠ CDA are right angles as they are congruent and supplementary.

Therefore, □ ABCD is a rectangle since a

, □ ABCD is a rectangle since a parallelogram with one right interior angle is a rectangle.