Question

In a triangle ABC, if D is midpoint of BC; AD is produced upto E such as DE = AD, then prove that:
a. DABD andDECD are congruent.
b. AB = EC
c. AB is parallel to EC

Answer 1

Given:
D os mid-point of BC
⇒ BD = DC
DE = AD
To prove:
a. ΔABD ≅ ΔECD
b. AB = EC
c. AB || EC

a. In ΔABD and ΔECD
BD = DC              ....(given)
∠ADB = ∠CDE   ....(vertically opposite angles)
AD = DE              ....(given)
∴ By Side-Angle-Side criterion of congruence,
ΔABD ≅ ΔECD

b. The corresponding parts of the congruent triangle are congruent.
∴ AB = EC

c. Also, ∠DAB = ∠DEC   ....(c.p.c.t)
∴ AB || EC                        ....(∠DAB and ∠DEC are alternate angles).