1. In the diagram, $\triangle A B C$ is equilateral. Point $D$ is inside $\triangle A B C$ so that $\triangle B D C$ is right-angled at $D$ and has $D B=D C$. If $\angle A B D=x^{\circ}$, what is the value of $x$ ?
1. Since $\triangle A B C$ is equilateral, then $\angle A B C=60^{\circ}$. ( $\triangle A B C$ has three equal angles whose measures add to $180^{\circ}$.)
Since $\triangle B D C$ is right-angled at $D$ and has $D B=D C$, then $\triangle B D C$ is a right-angled isosceles triangle, which makes $\angle D B C=45^{\circ}$. (Here, $\angle D B C=\angle D C B$ since $D B=D C$ and the measures of the two angles add to $90^{\circ}$, which makes each $45^{\circ}$.)
Therefore, $x^{\circ}=\angle A B D=\angle A B C-\angle D B C=60^{\circ}-45^{\circ}=15^{\circ}$.
Thus, $x=15$.
ANSWER: $x=15$