Full solution

33

. In the given diagram,

\triangle A B E \cong \triangle C B D

. Prove:

\triangle A F D \cong \triangle C F E

Identify Given Information: Identify given information and the goal. $\newline$ Given: Triangles $ABE$ and $CBD$ are similar, which means their corresponding angles are equal. $\newline$ Goal: To prove that triangles $AFD$ and $CFE$ are similar.
Use Similar Triangles: Use the fact that similar triangles have equal corresponding angles. $\newline$ Since triangles $ABE$ and $CBD$ are similar, we have: $\newline$ $\angle ABE = \angle CBD$ (by the definition of similar triangles).
Identify Equal Angles: Identify other equal angles due to the geometry of the diagram. $\newline$ In triangle $AFD$ and triangle $CFE$ , we have: $\newline$ $\angle AFD$ and $\angle CFE$ are vertical angles, so they are equal.
Use Transitive Property: Use the transitive property of equality to relate angles in different triangles. $\newline$ Since $\angle ABE = \angle CBD$ and $\angle ABE$ is also equal to $\angle AFD$ (as they are the same angle), we can say that $\angle AFD = \angle CBD$ .
Identify Third Pair of Angles: Identify the third pair of equal angles using the fact that the sum of angles in a triangle is $180$ degrees. $\newline$ In triangle AFD, the sum of angles is $\angle AFD + \angle ADF + \angle FDA = 180$ degrees. $\newline$ In triangle CFE, the sum of angles is $\angle CFE + \angle CEF + \angle FEC = 180$ degrees. $\newline$ Since we have $\angle AFD = \angle CFE$ and $\angle ADF = \angle CEF$ (as they are corresponding angles in similar triangles ABE and CBD), it follows that $\angle FDA = \angle FEC$ .
Conclude Similar Triangles: Conclude that triangles $AFD$ and $CFE$ are similar. Since we have shown that all corresponding angles are equal ( $\angle AFD = \angle CFE$ , $\angle ADF = \angle CEF$ , and $\angle FDA = \angle FEC$ ), by the Angle-Angle (AA) similarity postulate, triangles $AFD$ and $CFE$ are similar.