Question

In the given figure ABCD is a parallelogram, AB is Produced to L and E is a midpoint of BC. Show that:

a. DDCE ≅ DLDE
b. AB = BL
c. DC = `"AL"/(2)`

Answer 1

Given:
ABCD is a parallelogram, where BE = CE
To prove:
a. DDCE ≅ DLDE
b. AB = BL
c. DC = `"AL"/(2)`

a. In ΔDCE and ΔLBE
∠DCE = ∠EBL    ....[DC || AB, alternate angles]
CE = BE              ....[given]
∠DEC = ∠LEB   ....[vertically opposite angles]
∴ By Angle-Side-Angle criterion of congruence,
ΔDCE ≅ ΔLBE
The corresponding parts of the congruent triangles are congruent.
∴ DC = LB    ....(1)

b. DC = AB   ....(2)[opposite sides of a parallelogram]
From (1) and (2),
AB = BL        ....(3)

c. Al = AB + BL
⇒ AL = Ab + AB   ....[From (3)]
⇒ AL = 2AB   
⇒ Al = 2DC.         ....[From (2)]