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state and prove that converse to third side mid-Point theorem |
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Answer» The converse of midpoint theorem states that: "If a line segment is drawnpassingthrough the midpoint of any one side of a triangle andparallelto another side, then this line segmentbisectsthe remaining third side. Given in the figure A : AP=PB, AQ=QC. To prove: PQ || BC and PQ=1/2 BC Plan: To prove ▲ APQ ≅ ▲ QRC Proof steps: AQ=QC [midpoint] ∠ 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. thank you |
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