Competition · AMC preparation · step 4 of 4
AMC 8 · 2014 · #9
Grade 8 geometry-2d
Pick an answer.
AMC 8 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The problem is purely geometric, so Tool #1 (Draw a Diagram) is the starting move — sketch △ ABC, mark D on AC, draw BD, and tick the two equal sides BD and DC to make the isosceles structure of △ BDC visible. Once the picture is in place, Tool #7 (Identify Subproblems) splits the angle hunt into two clean steps: (a) find ∠ BDC inside the isosceles triangle △ BDC, then (b) use the fact that ∠ ADB and ∠ BDC form a straight line at D to get ∠ ADB. Each sub-step is a one-line angle calculation.
Spot the isosceles triangle
△ BDC has BD = DC, so it is isosceles with base BC; its base angles are equal, giving ∠ DBC = ∠ BCD = 70°.
Classifying △ BDC as isosceles from its equal side marks is the Grade 5 "classify 2D figures by properties" move.
5.G.B.4Draw A DiagramUse the 180 degree sum
Every triangle's interior angles sum to 180°, so ∠ BDC = 180° - 70° - 70° = 40°.
"The three interior angles of a triangle add to 180°" is the Grade 8 informal angle-sum fact.
8.G.A.5Identify SubproblemsTake the straight-line supplement
D lies on line AC, so ∠ ADB and ∠ BDC form a straight line and are supplementary: ∠ ADB = 180° - 40° = 140°.
Recognizing a linear pair (supplementary adjacent angles on a straight line) is the Grade 7 angle-relationship skill.
Because D lies on segment AC, ∠ ADB is the supplement of ∠ BDC, the apex angle of the isosceles triangle △ BDC — so finding ∠ ADB comes down to finding ∠ BDC.
▸ Why?
Since D is a point on segment AC, the rays DA and DC leave D in exactly opposite directions and form one straight line; segment DB splits that line into ∠ ADB and ∠ BDC, so those two angles fill the straight line together.
▸ Why?
That leaves only ∠ BDC to pin down, and ∠ BDC is the third angle of △ BDC, whose other two angles can be read off.
▸ Why?
The three interior angles of △ BDC add to a straight angle, so ∠ BDC equals that straight angle minus the other two angles.
▸ Why?
Those other two angles, ∠ DBC and ∠ BCD, are equal to each other, and ∠ BCD is given as 70°, so each of them measures 70°.
▸ Why?
∠ DBC and ∠ BCD are the base angles of △ BDC, which is isosceles because BD = DC; folding the triangle across the line through D perpendicular to base BC lays side DB onto side DC and carries ∠ DBC exactly onto ∠ BCD, so the two base angles are equal.
This AMC 8 problem only needs the Grade 8 fact that a triangle's three angles add to 180° — plus the Grade 7 fact that two angles on a straight line add to 180°.
- Spot the isosceles triangle
- Use the 180 degree sum
- Take the straight-line supplement
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