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.