If the non parallel sides of a trapezium are equal. Prove that it is cyclic.

Answer

To prove that a trapezium with equal non-parallel sides is cyclic, we can follow these steps:

### Step 1: Define the trapezium
Let trapezium ABCD have parallel sides AB and CD, and non-parallel sides AD and BC. Given that AD = BC, we know that the non-parallel sides are equal.

### Step 2: Identify the angles
Since AD = BC, by the properties of isosceles trapeziums, we have:
- Angle A = Angle B (the angles adjacent to one parallel side are equal)
- Angle C = Angle D (the angles adjacent to the other parallel side are equal)

### Step 3: Express the sum of angles in the trapezium
In any quadrilateral, the sum of the interior angles is 360 degrees. Therefore, we can write:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} + \text{Angle D} = 360^\circ \]

### Step 4: Substitute the equal angles
Since Angle A = Angle B and Angle C = Angle D, we can substitute:
Let Angle A = Angle B = x and Angle C = Angle D = y. Thus, we have:
\[ x + x + y + y = 360^\circ \]
This simplifies to:
\[ 2x + 2y = 360^\circ \]

### Step 5: Simplify the equation
Dividing the entire equation by 2 gives:
\[ x + y = 180^\circ \]

### Step 6: Identify the opposite angles
Here, Angle A and Angle C are opposite angles, and we have shown that:
\[ \text{Angle A} + \text{Angle C} = 180^\circ \]
Similarly, since Angle B = Angle A and Angle D = Angle C, we also find:
\[ \text{Angle B} + \text{Angle D} = 180^\circ \]

### Conclusion
Since the sum of the opposite angles in trapezium ABCD is equal to 180 degrees, we conclude that trapezium ABCD is a cyclic quadrilateral.