Problem:
In the figure, ∠A,∠B and ∠C are right angles. If ∠AEB=40∘ and ∠BED=∠BDE, then ∠CDE=
Answer Choices:
A. 75∘
B. 80∘
C. 85∘
D. 90∘
E. 95∘
Solution:
In △BDE,∠BED+∠BDE+∠B=180∘. Since ∠BED=∠BDE and ∠B=90∘, it follows that ∠BED=∠BDE=45∘. In △AEF, ∠A+∠AEF+∠AFE=180∘. Since ∠A=90∘ and ∠AEF=40∘, it follows that ∠AFE=50∘. Consequently ∠BFG=50∘ in △BFG and, since ∠B= 90∘, it follows that ∠BGF=40∘. Consequently ∠CGD=40∘ in △CDG, and since ∠C=90∘, it follows that ∠CDG=50∘. Thus ∠CDE=50∘+45∘=95∘.
OR
As in the first solution, ∠BED=∠BDE=45∘. Then ∠AED=40∘+45∘= 85∘. Since the four angles of a quadrilateral sum to 360∘, we have ∠A+∠C+ ∠AED+∠CDE=360∘. Thus ∠CDE=360∘−90∘−90∘−85∘=95∘.
Answer: E​.
The problems on this page are the property of the MAA's American Mathematics Competitions
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