Suppose ABCD is a trapezium in which AB is parallel to CD and AD=BC.prove that angle A=angle B and angle C=angle D.
Consider the right triangles with AD and BC as hypotenuses. Their height is the same, so their base legs are also the same. So, with all three sides congruent, the angles are also congruent.
To prove that angle A is equal to angle B and angle C is equal to angle D in the given trapezium ABCD, you can use the properties of a trapezium and the fact that AD is equal to BC.
Here's the step-by-step proof:
1. Given ABCD is a trapezium with AB parallel to CD and AD=BC.
2. In a trapezium, the opposite angles are supplementary. Therefore, angle A + angle D = 180 degrees, and angle B + angle C = 180 degrees.
3. Since AD=BC (given), angles A and D are equal (corresponding angles) and angles B and C are equal (corresponding angles).
4. Substituting the equal values, we have angle A + angle A = 180 degrees and angle B + angle B = 180 degrees.
5. Simplifying, we get 2 * angle A = 180 degrees and 2 * angle B = 180 degrees.
6. Dividing both sides of the equations by 2, we have angle A = 90 degrees and angle B = 90 degrees.
7. Therefore, angle A is equal to angle B and angle C is equal to angle D, as required.
In conclusion, by using the properties of a trapezium and the fact that AD is equal to BC, we have proved that angle A is equal to angle B and angle C is equal to angle D in the given trapezium ABCD.