In the figure, $\overline{\mathrm{TI}}\perp \overline{\mathrm{IY}},$ $\overline{\mathrm{DY}}\perp \overline{\mathrm{YI}},$ and $\angle T\cong \angle D.$
Prove that $\overline{\mathrm{TI}}\cong \overline{\mathrm{DY}}.$

In the figure above, $\mathrm{\Delta TIY}$ and $\mathrm{\Delta DYI}$ are overlapping triangles. In cases of adjacent and overlapping triangles, look for segments or angles shared by the two triangles that can be used to prove that the two triangles are congruent.