Question

Prove that: In a parallelogram, opposite angles are equal.

Answer 1

Hint: We know that the opposite sides of the parallelogram is parallel. Now drawing a transversal line AC, in the parallelogram ABCD, the alternate angle becomes equal and we get two different equations. Adding these both equations, at last , we prove that the opposite angles of the parallelogram are equal.

Complete step-by-step answer:

It is already given in the question that a parallelogram ABCD has AC as its one of the diagonal.
To prove:- $\angle $A = $\angle $C and
  $\angle $B = $\angle $D.
Proof:- Opposite sides of parallelogram is parallel.
So, AB$\parallel $CD and AD$\parallel $BC.
 Since, AB$\parallel $CD;
 And AC is the transversal.
 $\angle $BAC = $\angle $DCA ( They are the Alternate angles)
So,$\angle $ BAC = $\angle $DCA …(1)
Now, Since, AD$\parallel $BC;
  And AC is the transversal.
 $\angle $DAC = $\angle $BCA (They are the Alternate angles)
 So, $\angle $DAC = $\angle $BCA ….(2)
Adding both the equations, that is equation (1) and (2) , we get;
$ \Rightarrow $$\angle $BAC + $\angle $DAC = $\angle $DCA + $\angle $BCA
$ \Rightarrow $ $\angle $BAD = $\angle $DCB.
$ \Rightarrow $ $\angle $A = $\angle $C.
In the similar way only;
We can prove that:- $\angle $ADC = $\angle $ABC
$ \Rightarrow $ $\angle $D = $\angle $ B.
Hence, it is proved that in a parallelogram , the opposite sides of parallelogram are equal.

Note: In order to solve this particular question, we need to memorize these properties of parallelogram:-
a.) The opposite sides are congruent.
b.) The opposite angles are congruent.
c.) The consecutive angles are supplementary.
d.) If anyone of the angles is a right angle, then all the other angles will be the right angle.
e.) The two diagonals bisect each other.