1996 AHSME Problems/Problem 21
Contents
Problem
Triangles and are isosceles with , and intersects at . If is perpendicular to , then is
Solution 1
Redraw the figure as a concave pentagon :
The angles of the pentagon will still sum to , regardless of whether the pentagon is concave or not. As a quick proof, note that the nine angles of three original triangles , , and all make up the angles of the pentagon without overlap.
Since reflex , we have:
.
From isosceles , we get , so:
From isosceles , we get , so:
, which is answer
Solution 2
Let . By the isosceles triangle theorem, we have and . Because the angles of a triangle sum to , we have , then . Then we have . Substituting, this becomes . Adding , which is , we have
See also
1996 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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