Proving the Converse of the Parallelogram Opposite Sides Theorem: A Step-by-Step Guide

Parallelogram Opposite Sides Converse

The converse of the Parallelogram Opposite Sides Theorem states that if the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram

The converse of the Parallelogram Opposite Sides Theorem states that if the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

To understand and prove this converse, let’s break it down step by step:

Given: A quadrilateral ABCD, where AB = CD and AD = BC.

To prove: ABCD is a parallelogram.

Proof:

Step 1: Draw line segments AC and BD to connect the opposite vertices of the quadrilateral ABCD.

Step 2: Assume ABCD is not a parallelogram. This means either the opposite sides are not parallel or any one pair of opposite sides are not congruent.

Step 3: If opposite sides are not parallel, then lines AC and BD will intersect at a point P inside the quadrilateral.

Step 4: Choose any two points, Q and R, on AC and BD respectively, such that P is between Q and R.

Step 5: Since ABCD is not a parallelogram, lines AB and CD are not parallel. Therefore, the lines AB and CD intersect at a point S.

Step 6: Consider the triangles APS and CPS. Since AC and BD are transversals, we have the following corresponding angles: ∠APQ ≅ ∠CPR (vertical angles) and ∠SPA ≅ ∠CSP (alternate interior angles). Also, ∠APQ + ∠SPA = ∠CPR + ∠CSP = 180 degrees (linear pair).

Step 7: By the Angle-Side-Angle (ASA) congruence criterion, we can conclude that triangle APS ≅ triangle CPS. This implies that AP = CP and AS = CS.

Step 8: Since AP = CP and AS = CS, we can conclude that quadrilateral APSC is a parallelogram, as opposite sides are congruent.

Step 9: Now, consider the triangles BPS and DPS. Using similar reasoning as Step 6 and Step 7, we can prove that triangle BPS ≅ triangle DPS, and consequently, BP = DP and BS = DS.

Step 10: Therefore, quadrilateral BPDS is also a parallelogram.

Step 11: From quadrilateral APSC and quadrilateral BPDS, we notice that their opposite sides AP = CS and BP = DS are congruent.

Step 12: By the Parallelogram Opposite Sides Theorem, as opposite sides are congruent, a quadrilateral is a parallelogram. Thus, ABCD must be a parallelogram.

Step 13: This contradicts our initial assumption that ABCD is not a parallelogram. Therefore, our assumption was incorrect, and ABCD is indeed a parallelogram.

Hence, we have successfully proved the converse of the Parallelogram Opposite Sides Theorem, which states that if the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

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