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# 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 = 90o (Property of perpendicular lines)
Since CF is perpendicular to BP (Given), then angle CFD = 90o (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.