# 1993 AHSME Problems/Problem 2

## Problem

$[asy] draw((-5,0)--(5,0)--(2,14)--cycle,black+linewidth(.75)); draw((-2.25,5.5)--(4,14/3),black+linewidth(.75)); MP("A",(-5,0),S);MP("C",(5,0),S);MP("B",(2,14),N);MP("E",(4,14/3),E);MP("D",(-2.25,5.5),W); MP("55^\circ",(-4.5,0),NE);MP("75^\circ",(5,0),NW); [/asy]$

In $\triangle ABC$, $\angle A=55^\circ$, $\angle C=75^\circ, D$ is on side $\overline{AB}$ and $E$ is on side $\overline{BC}$. If $DB=BE$, then $\angle{BED} =$

$\text{(A) } 50^\circ\quad \text{(B) } 55^\circ\quad \text{(C) } 60^\circ\quad \text{(D) } 65^\circ\quad \text{(E) } 70^\circ$

## Solution

We first consider $\angle CBA$. Because $\angle A = 55$ and $\angle C = 75$, $\angle B = 180 - 55 - 75 = 50$. Then, because $\triangle BED$ is isosceles, we have the equation $2 \angle BED + 50 = 180$. Solving this equation gives us $\angle BED = 65 \rightarrow \fbox{\textbf{(D)}65}$

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