Question

In the diagram of PARALLELOGRAM ABCD, segment BE is perpendicular to segment CED, segment DF is perpendicular to BFC and segment CE is congruent to segment CF.
Prove ABCD is a rhombus.​

Answer 1

Answer:

In parallelogram ABCD

FD is perpendicular to BC

BE is perpendicular to CD

Consider triangle BEC and triangle DFC

FC = EC (Given)

Angle BEC = Angle DFC (=90°)

Angle BCE = Angle DCF (common)

Therefore triangle BEC is congruent to triangle DFC (AAS congruency)

DF = BE (CPCT)

Since the altitudes are equal their bases will also be equal

Therefore BC = DC

Therefore BC = DC = AD = AB

Therefore ABCD is a rhombus

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