Problem:
In the figure, β A,β B and β C are right angles. If β AEB=40β and β BED=β BDE, then β CDE=
Answer Choices:
A. 75β
B. 80β
C. 85β
D. 90β
E. 95β
Solution:
In β³BDE,β BED+β BDE+β B=180β. Since β BED=β BDE and β B=90β, it follows that β BED=β BDE=45β. In β³AEF, β A+β AEF+β AFE=180β. Since β A=90β and β AEF=40β, it follows that β AFE=50β. Consequently β BFG=50β in β³BFG and, since β B= 90β, it follows that β BGF=40β. Consequently β CGD=40β in β³CDG, and since β C=90β, it follows that β CDG=50β. Thus β CDE=50β+45β=95β.
OR
As in the first solution, β BED=β BDE=45β. Then β AED=40β+45β= 85β. Since the four angles of a quadrilateral sum to 360β, we have β A+β C+ β AED+β CDE=360β. Thus β CDE=360ββ90ββ90ββ85β=95β.
Answer: Eβ.
The problems on this page are the property of the MAA's American Mathematics Competitions