Question

ABCD is a parallelogram. AD is produced to E so that $ DE=DC $ and EC produced meets AB produced in F. Prove that $ BF=BC $ .

Answer 1

Hint: First, here we will prove that \[\angle E=\angle C\] using the concept that equal sides have opposite angles equals. Then, we will prove that corresponding angles formed by line AB and CD are equal. Corresponding angles is when two lines are crossed by another line i.e. called as transversal, the angles in matching corners are called corresponding angles. Then, using this \[AD+DE=AB+BF\] we will prove $ BF=BC $ . Diagram will be as shown below:

Complete step-by-step answer:
Here, we will first draw the figure as per the data given to us. So, we will get figure as

It is given that $ DE=DC $ . We have to prove $ BF=BC $ .
We will take $ \Delta DEC $ where $ DE=DC $ . So, we know that equal sides have opposite angles equals. So, angle opposite to side DC i.e. $ \angle DEC $ is equal to $ \angle DCE $ which is the opposite angle of side DE.
We can write it as $ \angle DEC=\angle DCE $ …………………(1)
We know that here sides $ AB\parallel CD $ , then corresponding angle formed by these parallel lines are also equal i.e. given as when two lines are crossed by another line i.e. called as transversal, the angles in matching corners.
 So, we can write it as
 $ \angle DCE=\angle BFC $ ……………………….(2)
So, from equation (1) we will substitute value of $ \angle DCE $ in equation (2) we will get as
 $ \angle DEC=\angle BFC $ …………………….(3)
Now, we will take \[\Delta AEF\] where we know that angle E and angle F are equal from equation (3). So, we can write it as
\[AE=AF\] (using the concept equal sides have opposite angles equals)
So, we can further write this as
\[AD+DE=AB+BF\]
Here, we can say that \[AB=CD=DE\] from the figure. So, we get equation as
\[AD=BF\]
But we know that \[AD=BC\] . So, on putting this value we get as
 $ BC=BF $ ……………..(4)
Thus, proved.

Note: In this type of proof that problems, diagrams are needed otherwise problems are not solved. Students should know all the concepts regarding this i.e. corresponding angle and what are the rules for the angle to be equal with each other. Then only it will be easy to solve. So, be clear with the concept and then solve it.