ABCD is a cyclic trapezium whose sides AD and BC are parallel to each other .If angle ABC = 72 degree then what is the measure of the angle BCD .
In a cylic quad, opposite angles are supplementary, so
angle D = 108°
but since AD || BC, angle C + angle D i= 180° as well
so angle C = 72°
To find the measure of angle BCD in the given cyclic trapezium ABCD, we can make use of the property that opposite angles in a cyclic quadrilateral are supplementary.
Let's follow these steps to find the measure of angle BCD:
1. Recall that a cyclic quadrilateral is a quadrilateral where all its vertices lie on a common circle.
2. In the given problem, ABCD is a cyclic trapezium, which means that all four vertices - A, B, C, and D - lie on a common circle.
3. Since AD and BC are parallel, it implies that angle ACD and angle BCD are corresponding angles and thus congruent. Therefore, to find the measure of angle BCD, we need to calculate the measure of angle ACD first.
4. We know that opposite angles in a cyclic quadrilateral are supplementary. The opposite angle to angle ABC is angle ADC.
5. Since angle ABC is given as 72 degrees, we can deduce that angle ADC (opposite angle to angle ABC) will also be 72 degrees.
6. Now, we have angle ADC = 72 degrees.
7. By the property of trapezium, we know that angle ACD + angle ADC = 180 degrees.
8. Substituting the known values, we get angle ACD + 72 = 180.
9. Solving for angle ACD, we have angle ACD = 180 - 72.
10. Therefore, angle ACD = 108 degrees.
11. Since angle ACD and angle BCD are congruent, we conclude that angle BCD = 108 degrees.
Hence, the measure of angle BCD in the given cyclic trapezium ABCD is 108 degrees.