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# Triangles: Isosceles

In the figure below, what is the value of $$x$$? ![](data:image/svg+xml;base64,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)
Incorrect. [[snippet]]
Incorrect. [[snippet]]
Correct. [[snippet]] Given that $$AD = DB = DC$$, angles $$CAD$$, $$ACD$$, $$DCB$$, and $$DBC$$ are equal. Since the sum of all angles in a triangle is 180° and $$DCB = BCD$$, >$$DCB = \frac{180^\circ - 90^\circ}{2} = 45^\circ$$. Likewise, angle $$ACD = 45^\circ$$. Based on this, angle $$ACB = 90^\circ$$ and so is its supplementary angle $$ACE$$. Hence, $$x = 90^\circ$$.
Incorrect. [[snippet]]
Incorrect. [[snippet]]
60°
75°
90°
105°
135°