Problem 2 · The kite and the 20° angle
Exterior angles and a hidden isosceles triangle: $CD = AD = EA$.
Integer answer, at most 4 digitsThe quadrilateral $ABCD$ is a kite, i.e. the line through $B$ and $D$ is an axis of symmetry of the quadrilateral. Extending sides $AB$ and $CD$, they form an angle of $20°$, and the angle $\angle DAB$ is $40°$. If segment $EA$ measures 2026 cm, how long is side $CD$ of the kite?
Reasoned solution
Key idea: in the figure, $E$, $A$ and $B$ are collinear, with $A$ between $E$ and $B$, and lines $EB$ and $EC$ meet at $E$ at an angle of $20°$.
In triangle $EAD$: the angle at $E$ is $20°$ and $\angle DAE = 180° - \angle DAB = 180° - 40° = 140°$ (supplementary angles on line $EB$). Hence
Triangle $EAD$ has two $20°$ angles (at $E$ and at $D$): it is isosceles with $AD = EA = 2026$ cm.
By the kite's symmetry about line $BD$, side $CD$ is the mirror image of side $AD$: