AMC 10 · 2008 · #24

Grade 8 geometry-2d

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

isosceles-triangleequilateral-triangleangle-sum-triangle identify-subproblems ↑ Prerequisites: isosceles-triangle

📏 Long solution 💡 4 insights

Problem

Quadrilateral ABCD has three equal consecutive sides AB = BC = CD. The corner at B measures 70 degrees and the corner at C measures 170 degrees. Find the measure of angle BAD, the corner at A.

Pick an answer.

(A)

$\ 75$

(B)

$\ 80$

(C)

$\ 85$

(D)

$\ 90$

(E)

$\ 95$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

Three sides share the same length, which begs for an auxiliary construction that recycles that length. Building an equilateral triangle on CD manufactures a fourth segment of the same length plus a clean 60 degree angle. That 60 degree slice tames the 170 degree corner into 110 degrees, and the figure splits into two friendly pieces: a parallelogram and an isosceles triangle. Then angle BAD is just one corner minus a smaller one.

1STEP 1

Build an equilateral triangle on CD

Let AB = BC = CD = 1 and build equilateral triangle CDE on CD, E on B's side: CE = DE = 1 and its angles are 60 degrees.

CE = DE = CD = 1, ∠ DCE=∠ CDE=∠ DEC=60°

2STEP 2

Slice the 170 degree corner

Ray CE peels the 60 degree wedge off the corner at C, leaving angle BCE = 170 - 60 = 110 degrees, and CE = CB = 1.

∠ BCE=∠ BCD-∠ DCE=170°-60°=110°, CE=CB=1

3STEP 3

Spot a pair of parallel sides

Across transversal BC the same-side angles give 70 + 110 = 180 degrees, so BA is parallel to CE and ABCE is a parallelogram.

∠ ABC+∠ BCE=70°+110°=180° → BA ∥ CE, AB=CE

4STEP 4

Cash in the parallelogram

Opposite parts of ABCE match: EA = BC = 1, angle EAB = 110 degrees, and angle AEC = 70 degrees.

EA=BC=1, ∠ EAB=∠ BCE=110°, ∠ AEC=∠ ABC=70°

5STEP 5

Read the isosceles triangle AED

Triangle AED has AE = ED = 1 with apex angle AED = 70 + 60 = 130, so each base angle is angle DAE = 25 degrees.

AE=ED=1, ∠ AED=∠ AEC+∠ CED=70°+60°=130°, ∠ DAE=(180°-130°)/2=25°

6STEP 6

Subtract to reach angle BAD

Ray AD lies inside corner EAB, so angle BAD = 110 - 25 = 85 degrees.

∠ BAD=∠ EAB-∠ DAE=110°-25°=85°

Answer

Check the four corners of ABCD. We found angle BAD = 85 degrees. The corner at D is angle ADC = angle CDE - angle ADE = 60 - 25 = 35 degrees. Then 85 + 70 + 170 + 35 = 360 degrees, exactly the angle sum a quadrilateral must have, so the answer is consistent. It is also comfortably below 120 degrees, the total the two unknown corners at A and D must share (360 - 70 - 170), which rules out any value that would leave too little for D.

💡Key takeaway

When a shape has several equal sides, build an equilateral triangle on one of them: it copies that length and drops a clean 60 degree angle that can turn a messy corner into parallel lines you can actually use.

Build an equilateral triangle on CD
Slice the 170 degree corner
Spot a pair of parallel sides
Cash in the parallelogram
Read the isosceles triangle AED
Subtract to reach angle BAD