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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 = 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.