Quadrilateral with equal opposite angle is a parallelogram

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

quadrilateral with equal opposite angle is a parallelogram

Consider the quadrilateral ABCD in which opposite angles are equal.
∠A = ∠C
∠B = ∠D

To prove:
Prove that ABCD is a parallelogram.

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.

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