2. Given the triangle $A D C$, segment $B D$ is a perpendicular bisector of segment $A C$. Point $B$ is the median of side $A C$. Why would triangle $A B D$ be similar to triangle $C B D$ ?
Answer
\(\boxed{\text{Triangle ABD is similar to triangle CBD because they have the same shape, their corresponding angles are equal (angle ABD = angle CBD = 90 degrees), and their corresponding sides are in proportion (AB/BC = BD/BD = 1).}}\)
Step 1 ：Given the triangle $A D C$, segment $B D$ is a perpendicular bisector of segment $A C$. Point $B$ is the median of side $A C$.
Step 2 ：In order for two triangles to be similar, they must have the same shape, but not necessarily the same size. This means that their corresponding angles must be equal and their corresponding sides must be in proportion.
Step 3 ：Since BD is a perpendicular bisector of AC, it means that angle ABD is equal to angle CBD (both are 90 degrees).
Step 4 ：Also, since B is the median of AC, it means that AB is equal to BC. Therefore, the corresponding sides AB/BC = BD/BD = 1, which means the sides are in proportion.
Step 5 ：Based on the above, we can conclude that triangle ABD is similar to triangle CBD.
Step 6 ：\(\boxed{\text{Triangle ABD is similar to triangle CBD because they have the same shape, their corresponding angles are equal (angle ABD = angle CBD = 90 degrees), and their corresponding sides are in proportion (AB/BC = BD/BD = 1).}}\)
