Question

In the above figure, ACDF, ABEF and BCDE are all parallelograms. Analyse the given figure and choose the correct option.

• A. Opposite sides are equal.
• B. Opposite angles are equal.
• C. Opposite angles are bisected by the diagonals.
• D. Diagonals bisect each other.

Answer 1

The correct option is C

Opposite angles are bisected by the diagonals.

Since ABEF is a parallelogram, so AF || BE and BF is the transversal.
$\therefore \angle$ AFB = $\angle$ FBE = x (alternate angles)..............(1)
Also, since BCDE is a parallelogram,
BE || CD and BD is the transversal
$\therefore \angle$ CDB = $\angle$ EBD = y (alternate angles) .............(2)
Now,
$\angle$FBD = $\angle$FBE + $\angle$ EBD
= x + y
And, $\angle$AFE = $\angle$BED (corresponding angles)
$\therefore$ $\angle$AFE = z
$\angle$ BED = $\angle$ BCD = z (opposite angles of a parallelogram)
$\angle$ CDG + $\angle$ BDC = $\angle$ EBD + $\angle$ BED (sum of internal opposite angles equals external angles of a $\Delta$)
$\Rightarrow \angle$CDG = $\angle$ EBD + $\angle$BED - $\angle$ BDC
= y + z - y
= z