ABCD is a cyclic trapezium in which side AD is parallel to side BC. If angle ABC is 70 degrees, then what is the measure of angle BCD in degrees?

Introduction / Context: This problem combines properties of cyclic quadrilaterals and trapeziums. It checks your understanding of angle relationships when a quadrilateral is both cyclic and a trapezium with a pair of parallel sides. Recognizing how opposite and adjacent angles relate in such special figures is important for solving geometry problems efficiently in competitive exams.

Given Data / Assumptions:

• ABCD is a cyclic quadrilateral, that is, all four vertices lie on a circle.
• ABCD is also a trapezium with AD parallel to BC.
• Angle ABC = 70 degrees.
• We must find angle BCD (the interior angle at C).
• Angles are measured in degrees in standard Euclidean geometry.

Concept / Approach: Two key concepts are used:

• In a cyclic quadrilateral, the sum of a pair of opposite interior angles is 180 degrees.
• In a trapezium, if a pair of opposite sides are parallel, the interior angles on the same side of a transversal sum to 180 degrees.

With AD parallel to BC, consider the leg AB as a transversal. The angles at A and B on this transversal are supplementary. At the same time, in a cyclic quadrilateral, angle A and angle C are supplementary. Combining these facts will lead us to a relationship between angle B and angle C.

Step-by-Step Solution: Step 1: Since AD is parallel to BC and AB is a transversal, angles DAB and ABC are interior angles on the same side of AB. Step 2: Therefore, angle DAB + angle ABC = 180 degrees. Step 3: Let angle DAB = A and angle BCD = C. We are given angle ABC = 70 degrees. Step 4: From Step 2, A + 70 = 180, so A = 110 degrees. Step 5: In a cyclic quadrilateral, opposite interior angles sum to 180 degrees. Thus angle A + angle C = 180 degrees. Step 6: Substitute A = 110 degrees into A + C = 180: 110 + C = 180. Step 7: Solve for C: C = 180 - 110 = 70 degrees. Step 8: Therefore, angle BCD = 70 degrees.

Verification / Alternative check: We have found that angle A = 110 degrees and angle C = 70 degrees. Opposite angles in the cyclic quadrilateral are A and C, and B and D. A + C = 110 + 70 = 180 degrees, satisfying the cyclic condition. Similarly, B = 70 degrees and D = 110 degrees must be supplementary, which is consistent with AD parallel to BC and CD as a transversal. Thus all conditions of both a cyclic quadrilateral and trapezium are satisfied.

Why Other Options Are Wrong: 110°: This would make angle A also 110°, causing A + C to exceed 180° and breaking the cyclic quadrilateral property. 80° and 60°: These do not satisfy the requirement that opposite angles sum to 180° together with the parallel line condition. 90°: This is a typical distractor but does not work with the 70 degree angle and the cyclic plus parallel constraints.

Common Pitfalls: A common error is to assume that opposite angles in a trapezium are equal rather than supplementary, or to forget which pairs of angles are opposite in a cyclic quadrilateral. Another mistake is mixing up adjacent and opposite angles when applying the 180 degree sum. Drawing a clear diagram with all labels and angle marks can greatly reduce confusion and help in applying both sets of properties correctly.

Final Answer: The measure of angle BCD is 70°.