AMC 8 · 2000 · #24

Grade 7 geometry-2d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

angle-sum-triangleisosceles-trianglesupplementary-angles identify-subproblemscasework ↑ Prerequisites: angle-sum-triangleisosceles-triangle

📏 Medium solution 💡 4 insights 📊 Diagram

Problem

In the figure, points A, G, F form a triangle with ∠ A = 20° and equal base angles ∠ AFG = ∠ AGF. Lines through F also create triangle BFD. Find ∠ B + ∠ D.

Pick an answer.

(A)

$48^\circ$

(B)

$60^\circ$

(C)

$72^\circ$

(D)

$80^\circ$

(E)

$90^\circ$

AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The figure looks busy, but the relevant information lives in just two triangles: AGF (where ∠ A = 20° and the base angles are equal) and BFD (whose interior angles include ∠ B and ∠ D). Tool #7 (Identify Subproblems) splits the figure along these two triangles, linked at point F. Tool #1 (Draw a Diagram) is the sidekick: mark the known 20° at A, label the matching base angles, then use the straight line AFD to carry that information from one triangle to the other. No algebra, just the triangle-angle-sum rule applied twice.

1STEP 1

Subproblem 1: triangle AGF sums to 180° with equal base angles, so each base angle ∠ AFG = 80°.

20° + x + x = 180° → 2x = 160° → x = 80°, so ∠ AFG = 80°

2STEP 2

A, F, D are collinear, so ∠ AFG and ∠ GFD form a linear pair: ∠ GFD = 180° - 80° = 100°.

∠ GFD = 180° - ∠ AFG = 180° - 80° = 100°

3STEP 3

G lies on ray FB, so rays FG and FB point the same way — thus ∠ BFD and ∠ GFD are the same angle: ∠ BFD = 100°.

∠ BFD = ∠ GFD = 100°

4STEP 4

Subproblem 2: the angle sum of triangle BFD gives ∠ B + ∠ D = 180° - 100° = 80°.

∠ B + ∠ D + ∠ BFD = 180° → ∠ B + ∠ D = 180° - 100° = 80° → (D)

Answer

80°

The two unknown angles ∠ B and ∠ D can each take many values (the figure doesn't fix them separately), yet the problem expects a single number — that's already a hint that only the sum is determined, exactly what the triangle-angle-sum rule provides once ∠ BFD is known. The arithmetic checks: 20 + 80 + 80 = 180 in triangle AGF, 80 + 100 = 180 on line AD, and ∠ B + ∠ D + 100 = 180 in triangle BFD, giving 80° — choice (D). The answer is also positive and less than 180°, as any pair of triangle angles must be.

💡Key takeaway

The figure looks tangled, but only two triangles do the work: AGF on top and BFD on the bottom, meeting at F. The isosceles top triangle pins ∠ AFG = 80°; the straight line through F flips that to ∠ BFD = 100°; and the triangle-angle-sum rule in BFD forces ∠ B + ∠ D = 80° — choice (D).

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