AMC 10 · 2011 · #17

Grade 8 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

inscribed-anglearc-measuresupplementary-anglesangle-sum-triangle identify-subproblems ↑ Prerequisites: inscribed-angle

📏 Long solution 💡 3 insights 📊 Diagram

Problem

Five points A, B, C, D, E lie on a circle in that order. The diameter EB is parallel to chord DC, and chord AB is parallel to chord ED. In triangle AEB, the angles at E and B are in the ratio ∠ AEB : ∠ ABE = 4 : 5. What is the degree measure of ∠ BCD?

Pick an answer.

(A)

120

(B)

125

(C)

130

(D)

135

(E)

140

AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

This is an angle-chase on a labeled circle, so Tool #1 (work the diagram) is primary: mark every angle the givens force, then follow them to ∠ BCD. Tool #4 names the ratio parts as 4x and 5x so the 4:5 split becomes one equation. Tool #7 breaks the chase into clean stages — first the right angle from the diameter, then the parallel-line angle, then the cyclic-quadrilateral angle. Tool #3 matches the final number to a choice.

1STEP 1

Diameter forces a right angle

A sits on the circle looking across the whole diameter EB, and such a corner is always square: ∠ EAB = 90°.

∠ EAB = 90°

2STEP 2

The other two angles add to 90

The three angles of △ AEB total 180° and one is the right angle, so ∠ AEB and ∠ ABE add to 90°.

∠ AEB + ∠ ABE = 180° - 90° = 90°

3STEP 3

Split 90 in the ratio 4 : 5

Set ∠ AEB = 4x and ∠ ABE = 5x; then 9x = 90°, so x = 10° and ∠ ABE = 50°.

4x + 5x = 90° → x = 10°, ∠ ABE = 5x = 50°

4STEP 4

Slide the angle along parallel chords

The diameter EB cuts the parallel chords AB and ED, so ∠ ABE and ∠ DEB are alternate interior angles: ∠ DEB = 50°.

AB ∥ ED → ∠ DEB = ∠ ABE = 50°

5STEP 5

Opposite angles of the inscribed quadrilateral

In the inscribed quadrilateral BCDE the opposite corners C and E are supplementary, so ∠ BCD = 180° - 50° = 130°.

∠ BCD = 180° - ∠ DEB = 180° - 50° = 130°

6STEP 6

Match to a choice

∠ BCD = 130° matches choice (C).

130° → (C)

Answer

130

The answer should be an obtuse angle: C sits on the short arc near the top while B and D are spread wide, so ∠ BCD looking back across them is clearly more than 90° — 130° fits. Cross-check with arcs: the inscribed angle ∠ BCD cuts off the far arc BD (through E and A). Since ∠ DEB = 50° is an inscribed angle on arc BD (the near side), the far arc measures 360° - 2(50°) = 260°, and half of that is 130° — the same value, so the answer is consistent. The given EB ∥ DC pins down exactly where C lands, but the opposite-angle rule fixes ∠ BCD for any C on that arc, which is why we did not need it directly.

💡Key takeaway

A corner that spans a diameter is a right angle, a ratio just splits an angle into equal shares, and opposite corners of a circle-inscribed quadrilateral add to 180° — chain those and ∠ BCD = 130°.

Diameter forces a right angle
The other two angles add to 90
Split 90 in the ratio 4 : 5
Slide the angle along parallel chords
Opposite angles of the inscribed quadrilateral
Match to a choice