ABCD is a cyclic quadrilateral in which AB is a diameter of the circumscribed circle. Diagonals AC and BD intersect at E. If ∠DBC = 35°, what is the measure (in degrees) of ∠AED?

Introduction / Context:
This problem uses several properties of cyclic quadrilaterals and inscribed angles. Since AB is a diameter, certain angles are right angles. You are given an angle at vertex B and must find the angle between the diagonals at the intersection point E inside the quadrilateral.

Given Data / Assumptions:
• ABCD is a cyclic quadrilateral, so all its vertices lie on a circle.
• AB is a diameter of the circle.
• Diagonals AC and BD intersect at E.
• ∠DBC = 35 degrees.
• We must find ∠AED.

Concept / Approach:
First use the fact that AB is a diameter. Any angle subtended by a diameter at the circle is a right angle, so ∠ACB and ∠ADB are 90 degrees. Angle ∠DBC is an inscribed angle that intercepts arc DC, so its measure is half the measure of that arc. One can find the measure of arc DC, and then use the relationship between the arcs and the angle between diagonals in a cyclic quadrilateral: the angle between the diagonals at E has measure equal to half the sum of the measures of the arcs intercepted by that angle and its vertical opposite.

Step-by-Step Solution:
Step 1: ∠DBC is an inscribed angle intercepting arc DC, so m(arc DC) = 2 × ∠DBC = 2 × 35° = 70°. Step 2: Since AB is a diameter, arc AB is a semicircle, so m(arc AB) = 180°. Step 3: The full circle has 360°, so m(arc AB) + m(arc BC) + m(arc CD) + m(arc DA) = 360°. Step 4: Substitute known values: 180° + (arc BC + arc AD) + 70° = 360°, so arc BC + arc AD = 360° − 250° = 110°. Step 5: The angle between diagonals ∠AED is equal to half the sum of the measures of arcs AD and BC intercepted by the angle and its vertical opposite. Step 6: Therefore ∠AED = (1/2)(m(arc AD) + m(arc BC)) = (1/2)(110°) = 55°.

Verification / Alternative check:
In cyclic quadrilaterals, there are several symmetric relationships between diagonals and arcs. Visualizing or sketching the circle with AB as a horizontal diameter and points C and D above can help confirm that arcs AD and BC are the relevant arcs for ∠AED. The arithmetic of arcs is consistent with the full 360 degrees.

Why Other Options Are Wrong:
• 35° is the given angle at B, not the angle between the diagonals.
• 45° does not correspond to any correct arc sum in this configuration.
• 90° would imply perpendicular diagonals at E, which is not supported by the given data.

Common Pitfalls:
Students may confuse which arcs correspond to which inscribed or interior angles, or may use arc AB and arc CD instead of arc AD and arc BC. Carefully identifying which arcs are intercepted by the chords that form the angle at E is key.

Final Answer:
The measure of ∠AED is 55°.