AMC 10 · 2007 · #6
Grade 8 geometry-2dPick an answer.
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.
Draw the figure and split the angle at A
The corner angle splits into a difference of two base angles.
Since D sits inside the triangle, ray AD splits ∠ BAC, so the target angle is just the big base angle minus the small one.
4.MD.C.7Draw A DiagramFind angle BAC from triangle ABC
The first apex angle gives a base angle of 70.
Equal sides face equal angles, so the leftover after the apex splits evenly between the two base angles.
Equal sides face equal angles, so whatever the apex leaves over splits evenly between the two base angles.
▸ Why?
A triangle with two equal sides is a mirror image of itself, so its two base angles have to match.
▸ Why?
The three angles of a triangle always add to a straight angle, so the apex fixes what is left for the base.
Find angle DAC from triangle ADC
The second gives a base angle of 20.
A wide apex angle of 140° leaves only a little for the base angles, so each is a small 20°.
8.G.A.5Identify SubproblemsSubtract to get angle BAD
Subtracting gives 50, choice (D).
With both base angles known, the answer is a single subtraction of the small angle from the big one.
4.MD.C.7Draw A DiagramIn 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