AMC 10 · 2007 · #8

Grade 8 geometry-2d

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

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

📏 Medium solution 💡 2 insights

Problem

Two isosceles triangles share the base AC: triangle ABC has AB=BC with apex angle ∠ ABC=40°, and triangle ADC has AD=DC with apex angle ∠ ADC=140°. Point D sits inside triangle ABC. Find the measure of ∠ BAD.

Pick an answer.

(A)

$\ 20$

(B)

$\ 30$

(C)

$\ 40$

(D)

$\ 50$

(E)

$\ 60$

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

Nothing here needs algebra — the whole problem is reading angles off a picture, so Tool #1 (Draw a Diagram) comes first: sketch both isosceles triangles on the shared base AC and place D inside triangle ABC. The picture reveals the key relationship — ray AD falls between rays AB and AC, so ∠ BAC splits into ∠ BAD and ∠ DAC. That turns the goal into 'find ∠ BAC and ∠ DAC separately, then subtract,' which is Tool #7 (Identify Subproblems). Each of those two angles is a base angle of one isosceles triangle, found from the triangle's angle sum.

1STEP 1

Draw the figure and split the angle at A

Sketch AC with apex B above and D inside. Ray AD falls between rays AB and AC, so ∠ BAD=∠ BAC-∠ DAC.

∠ BAD=∠ BAC-∠ DAC

2STEP 2

Find angle BAC from triangle ABC

The apex ∠ ABC=40° leaves 140° for the two base angles, and AB=BC makes them equal, so ∠ BAC=70°.

∠ BAC=(180°-40°)/2=70°

3STEP 3

Find angle DAC from triangle ADC

Same move in triangle ADC: the wide apex ∠ ADC=140° leaves only 40° to split evenly, so ∠ DAC=20°.

∠ DAC=(180°-140°)/2=20°

4STEP 4

Subtract to get angle BAD

Feed the two base angles into the picture's relationship: ∠ BAD=70°-20°=50°, which is choice (D).

∠ BAD=70°-20°=50° → (D)

Answer

The answer must be a positive angle smaller than ∠ BAC=70°, since ∠ DAC=20° is carved out of it — and 50° fits. Sanity-check the whole picture: at A the pieces 20°+50°=70°=∠ BAC; triangle ABC has 70°+70°+40°=180° and triangle ADC has 20°+20°+140°=180°. Everything closes up, so 50° is consistent.

💡Key takeaway

In an isosceles triangle the two base angles are equal, so split the leftover after the top angle in half; here that gives 70° and 20°, and their difference is the answer.

Draw the figure and split the angle at A
Find angle BAC from triangle ABC
Find angle DAC from triangle ADC
Subtract to get angle BAD

Draw the figure and split the angle at A

Find angle BAC from triangle ABC

Find angle DAC from triangle ADC

Subtract to get angle BAD

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