Two chords AB and CD of a circle with centre O intersect at point P inside the circle. If ∠APC = 40°, then what is the value (in degrees) of ∠AOC + ∠BOD?

Introduction / Context:
This question uses properties of angles formed by intersecting chords inside a circle and the relationship between central angles and their corresponding arcs. You are given an angle formed by two chords at an interior point and asked for the sum of the corresponding central angles at the centre of the circle.

Given Data / Assumptions:
• AB and CD are chords of a circle with centre O.
• These chords intersect at an interior point P.
• The measure of angle ∠APC is 40 degrees.
• ∠AOC and ∠BOD are central angles subtending arcs AC and BD respectively.

Concept / Approach:
A key theorem states that the measure of an angle formed by two chords intersecting inside a circle is equal to half the sum of the measures of the intercepted arcs. For angle ∠APC formed by chords PA and PC, the intercepted arcs are arc AC and arc BD. Central angles ∠AOC and ∠BOD correspond directly to these arcs, because the measure of a central angle equals the measure of its subtended arc.

Step-by-Step Solution:
Step 1: Let m(arc AC) be the measure of arc AC and m(arc BD) be the measure of arc BD. Step 2: For intersecting chords, the theorem gives ∠APC = (1/2)[m(arc AC) + m(arc BD)]. Step 3: Central angle ∠AOC subtends arc AC, so m(arc AC) = ∠AOC. Step 4: Central angle ∠BOD subtends arc BD, so m(arc BD) = ∠BOD. Step 5: Therefore ∠APC = (1/2)[∠AOC + ∠BOD]. Step 6: Given ∠APC = 40 degrees, so 40 = (1/2)[∠AOC + ∠BOD]. Step 7: Multiply both sides by 2: ∠AOC + ∠BOD = 80 degrees.

Verification / Alternative check:
Conceptually, the arcs AC and BD together account for a certain part of the full 360 degree circle. The interior angle at P is precisely half of the sum of these arcs. Since the relationship is exact and direct, there is no need for additional information such as lengths or other angles for verification.

Why Other Options Are Wrong:
• 50 degrees or 60 degrees would correspond to using an incorrect fraction or misreading the theorem as using half the difference of arcs instead of half the sum.
• 120 degrees would come from using an incorrect factor of 3/2 or misunderstanding the relation between the interior angle and central angles.

Common Pitfalls:
Students often confuse this result with the formula for the angle formed by two secants or two tangents outside the circle, which uses half the difference of arcs. Remember that for intersecting chords inside the circle, you always use half the sum of the intercepted arcs.

Final Answer:
The value of ∠AOC + ∠BOD is 80°.