To verify the mid-point theorem for a triangle.
The line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is congruent to one half of the third side.
Basic concepts and facts
Two lines are parallel if they do not meet at any point.
Two triangles are congruent if their corresponding angles and corresponding sides are equal.
Two triangles are similar if their corresponding angles equal and their corresponding sides are in proportion.
Proof of theorem:
Given in the figure A :
PQ || BC and PQ=1/2 BC
To prove ▲ APQ ≅ ▲ QRC
∠ APQ = ∠QRC [Corresponding angles for parallel lines cut by an transversal].
∠PBR=∠QRC=∠APQ [Corresponding angles for parallel lines cut by an transversal].
∠RQC=∠PAQ [When 2 pairs of corresponding angles are congruent in a triangle, the third pair is also congruent.]
Therefore , ▲APQ ≅ ▲QRC
AP=QR=PB and PQ=BR=RC.
Since midpoints are unique, and the lines connecting points are unique, the proposition is proven.