The question in the image asks you to prove that triangles $\triangle ACD \cong \triangle BCD$, given that $\triangle ACE \cong \triangle BCE$. The diagram illustrates two triangles sharing a common vertex $C$, with lines connecting points $A$, $B$, $C$, $D$, and $E$.

Steps to prove $\triangle ACD \cong \triangle BCD$:

1. Given: $\triangle ACE \cong \triangle BCE$
   From this, you can deduce that corresponding sides and angles of these triangles are congruent. This implies:

   • $AC \cong BC$ (corresponding sides of congruent triangles)
   • $\angle ACE \cong \angle BCE$ (corresponding angles)
   • $AE \cong BE$ (corresponding sides)

2. Segment $CD$ is common to both triangles $\triangle ACD$ and $\triangle BCD$.
   Therefore, $CD \cong CD$ by the reflexive property of congruence.

3. Angles $\angle ACD \cong \angle BCD$
   Since $\triangle ACE \cong \triangle BCE$, the angles $\angle ACE$ and $\angle BCE$ are congruent. The line segment $CD$ divides these angles into two equal parts, which implies $\angle ACD \cong \angle BCD$ by the corresponding angles property.

4. Side $AC \cong BC$
   This was already established from the congruence of $\triangle ACE$ and $\triangle BCE$.

5. Conclusion: $\triangle ACD \cong \triangle BCD$
   By the Side-Angle-Side (SAS) Congruence Postulate, since two sides and the included angle of $\triangle ACD$ are congruent to two sides and the included angle of $\triangle BCD$, the triangles are congruent.

Would you like more details on any of these steps, or have any specific questions?

Related Questions:

1. What is the significance of the reflexive property in triangle congruence?
2. Can the Angle-Side-Angle (ASA) theorem be used to prove congruence in this case? Why or why not?
3. How do corresponding angles help in proving congruence between two triangles?
4. What other postulates, apart from SAS, can be used to prove triangle congruence?
5. Why is it necessary to specify that quadrilateral properties are not allowed in this proof?

Tip:

Always start a congruence proof by identifying the given congruent triangles and matching their corresponding parts systematically.