AMC 10 · 2009 · #9

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
isosceles-triangleangle-sum-trianglesupplementary-angles convert-to-algebra ↑ Prerequisites: isosceles-triangle
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
Two segments BD and AE cross at C, so B, C, D lie on one straight line and A, C, E lie on another. Four segments are equal: AB=BC=CD=CE. Also ∠ A is 5/2 times ∠ B. Find the degree measure of ∠ D.

Pick an answer.

(A)
52.5
(B)
55
(C)
57.7
(D)
60
(E)
62.5

AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The two angles are tied together by the ratio ∠ A = 5/2∠ B, so Tool #4 (Introduce a Variable) names ∠ B = x and turns every other angle into an expression in x. Tool #1 (Draw a Diagram) reads the figure to see two isosceles triangles sharing the crossing point C, and that the base angles pair off. Tool #13 (Convert to Algebra) writes the triangle-angle-sum fact as one equation and solves it. Tool #3 (Eliminate Possibilities) matches the final degree value against the five choices.

1STEP 1

Name the angle and use the first isosceles triangle

Let ∠ B = x. Since AB=BC, the angles opposite those sides match, so ∠ A = ∠ BCA — and ∠ A = 5/2 x, hence ∠ BCA = 5/2 x.

∠ B = x, ∠ A = ∠ BCA = 5/2x
2STEP 2

Solve for the angles of triangle ABC

The angles of triangle ABC add to 180: 5/2 x + x + 5/2 x = 180, so 6x = 180 and x = 30. Then ∠ B = 30 and ∠ A = ∠ BCA = 75.

5/2x + x + 5/2x = 180 → 6x = 180 → x = 30, ∠ BCA = 75
3STEP 3

Carry the angle across the crossing point

BD and AE are straight lines crossing at C, so ∠ DCE and ∠ BCA are vertical angles and must be equal: ∠ DCE = 75.

∠ DCE = ∠ BCA = 75
4STEP 4

Use the second isosceles triangle to find ∠ D

Triangle CDE has CD=CE, so ∠ D = ∠ E; from 75 + 2∠ D = 180 we get ∠ D = 52.5, which is choice (A).

75 + 2∠ D = 180 → ∠ D = 105/2 = 52.5 → (A)
Answer
52.5
Check every angle. Triangle ABC: 75 + 30 + 75 = 180. The vertical angle at C carries 75 into triangle CDE, whose angles are 75 + 52.5 + 52.5 = 180. Both triangles close, and ∠ D = 52.5 is a base angle of a fairly 'wide' apex (75°), so it should be a bit less than half of 180 — 52.5 fits. It lands exactly on choice (A).
💡Key takeaway

Name one angle, use 'equal sides mean equal base angles' plus the 180° triangle rule, then hop the angle across the crossing point with vertical angles to solve the second triangle.

  • Name the angle and use the first isosceles triangle
  • Solve for the angles of triangle ABC
  • Carry the angle across the crossing point
  • Use the second isosceles triangle to find ∠ D

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