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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)`
Solution
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)]
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