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:- $\mathrm{\angle }$A = $\mathrm{\angle }$C and
  $\mathrm{\angle }$B = $\mathrm{\angle }$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.
 $\mathrm{\angle }$BAC = $\mathrm{\angle }$DCA ( They are the Alternate angles)
So,$\mathrm{\angle }$ BAC = $\mathrm{\angle }$DCA …(1)
Now, Since, AD$\parallel$BC;
  And AC is the transversal.
 $\mathrm{\angle }$DAC = $\mathrm{\angle }$BCA (They are the Alternate angles)
 So, $\mathrm{\angle }$DAC = $\mathrm{\angle }$BCA ….(2)
Adding both the equations, that is equation (1) and (2) , we get;
$\Rightarrow$$\mathrm{\angle }$BAC + $\mathrm{\angle }$DAC = $\mathrm{\angle }$DCA + $\mathrm{\angle }$BCA
$\Rightarrow$ $\mathrm{\angle }$BAD = $\mathrm{\angle }$DCB.
$\Rightarrow$ $\mathrm{\angle }$A = $\mathrm{\angle }$C.
In the similar way only;
We can prove that:- $\mathrm{\angle }$ADC = $\mathrm{\angle }$ABC
$\Rightarrow$ $\mathrm{\angle }$D = $\mathrm{\angle }$ 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.