Theorem 3: The opposite angles of a parallelogram are equal.

GIVEN  A parallelogram ABCD

To prove  <A = <C and <B = <D

Proof Since ABCD is a parallelogram. Therefore,   AB DC and AD BC

Now, AB DC and transversal AD intersects them at A and D respectively.

   <A + <D = 180  ....(i)   [ Sum of consecutive interior angles is 180]

Again, AD BC and DC intersects them at D and C respectively.

   <C + <D = 180  ...(ii)     [ Sum of consecutive interior angles is 180]

From (i) and (ii), we get

      <A + <D = <D + <C

  <A = <C

Similarly, <B = <D.

Hence, <A = <C and <B = <D

Given : A quadrilateral ABCD in which and

To Prove: ABCD is a Parallelogram.

Proof: In a quadrilateral ABCD

                    .......(1)      [Given]

                 .......(2)      [Given]

Adding (1) and (2)

           .............(3)

Now            [Angle Sum Property]

Using equation (3) we get

But  and  are cointerior angles when AD and BC are straight lines and AB is the transversal cutting them

As their sum is   Therefore AD || BC         .......(4)

Again and  is given

But  and  are cointerior angles when AB and CD are straight lines and BC is the transversal cutting them

As their sum is   Therefore AB || CD                 .......(5)

From (4) and (5)

AD || BC and  AB || CD

Hence ABCD is a a parallelogram

Illustration: Find all the angles of the parallelogram ABCD the figure given.

Solution: In triangle BCD

            {angle sum property of triangle.]

5x+ 20+ 2x+ 10 +3x= 180

10x+ 30 = 180

10x = 180-30=150

x= 15

In a parallelogram sum of adjacent angles= 180

In a parallelogram opposite angles are equal

     and

Hence