# Congruent Triangles Example

- Question:
- In the figure above, ABC is a triangle. D is the midpoint of AC, connect points B and D, extend BD to point P. Draw a line passing through
point A and perpendicular to BD at point E. Draw another line passing through point C and perpendicular to BP at point F. Prove AE = CF.

- Proof:
- Since D is the midpoint of AC (Given), then AD = CD (Property of midpoint)
- Since AE is perpendicular to BD (Given), then angle AED = 90
^{o} (Property of perpendicular lines)
- Since CF is perpendicular to BP (Given), then angle CFD = 90
^{o} (Same reason)
- In triangle AED and triangle CFD,
- since angle AED = angle CFD, angle EDA = angle FDC (Vertical angles are equal), AD = CD
- So triangle AED is congruent to triangle CFD (AAS Theorem)
- So AE = CF (In two congruent triangles, corresponding sides are equal.)

- Note1:
- Since angle EDA = angle FDC, so angle EDA and angle FDC are a pair of corresponding angles. Since AE opposite angle EDA and CF opposite
angle FDC, so AE and CF are a pair of corresponding sides.

- Note2:
- Since BD is drawn from the vertex B to the midpoint of its opposite side AC, so BD is a median in the triangle ABC.

- What does this example tell us?
- The distances drawn from two endpoints of a side in a triangle to its median are equal.

- Use this example to help middle school students learning inference.