In this chapter we will prove that any quadrilateral with equal pair of opposite angles is actually a parallelogram.

Consider the quadrilateral ABCD in which opposite angles are equal.

∠A = ∠C

∠B = ∠D

**To prove:**

Prove that ABCD is a parallelogram.

**Solution:**

We know that **sum of all angles of quadrilateral measure 360 degree**.

∠A + ∠B + ∠C + ∠D = 360

Since, ∠A = ∠C and ∠B = ∠D, the above expression can be written as;

∠A + ∠B + ∠A + ∠B = 360

2 (∠A + ∠B) = 360

**∠A + ∠B = 180 degree **

The above expression can also be written as;

**∠C + ∠B = 180 degree **

Now let’s analyze each of the above equation.

∠A + ∠B = 180 degree

Both ∠A and ∠B are interior angles.

Their sum equal to 180 degree is only possible when AD & BC are parallel lines intersected by transversal CD.

Hence **AD || BC**.

We also know that ∠C + ∠B = 180 degree.

Again both ∠C & ∠B are interior angles.

Their sum equals to 180 degree mean that AB & CD are parallel line intersected by transversal BC.

Hence, **AB || CD**

After combining all the above expression we found that in above quadrilateral opposite sides are parallel to each other.

AB || CD and AD || BC

Hence, the given quadrilateral is a parallelogram.