1994 AMC 8 Problem 7

Attempt Problem 7 of the 1994 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1994 AMC 8 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

7.

If ∠A=60°,\angle A = 60°, ∠E=40°,\angle E = 40°, and ∠C=30°,\angle C = 30°, then ∠BDC=\angle BDC =
40°40°
50°50°
60°60°
70°70°
80°80°
Answer: B
Concepts:angle chasingangle sum
Small hint:
In triangle ABEABE, the angles at AA and EE determine the angle ∠ABE\angle ABE
Big hint:
Since DD lies on EBEB and A,B,CA, B, C are collinear, ∠DBC\angle DBC is the supplement of ∠ABE\angle ABE; then use triangle BDCBDC
Solution:
In triangle ABE,ABE, ∠ABE=180°\angle ABE = 180° −(60°+40°)- (60° + 40°) =80°.= 80°.
Since A,B,CA, B, C are collinear and DD lies on segment EB,EB, the angle ∠DBC=180°−80°=100°.\angle DBC = 180° - 80° = 100°.
In triangle BDC,BDC, ∠BDC=180°\angle BDC = 180° −(100°+30°)- (100° + 30°) =50°.= 50°.
Thus, the correct answer is B .
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