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Parallelogram proofs worksheets

The parallelogram theorems are where a geometry student first strings the earlier pieces together: alternate interior angles, the reflexive property, a congruence criterion, then CPCTC. These free printable worksheets give the figure, the Given and the Prove, and leave the table blank.

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Parallelogram proofs
Parallelogram proofs worksheet — Properties and converses, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

Every proof of a parallelogram property runs the same road, and seeing that once is worth more than any single proof: draw a diagonal to make two triangles, use the parallel sides to find equal angles, notice the diagonal is shared, close with ASA or SSS, then read off the part you were asked for. The sheets cover the properties the standard names — opposite sides congruent, opposite angles congruent, and the diagonals bisecting each other.

Why this sheet works

The second setting is the CONVERSE, and it is the harder half. There the quadrilateral is not yet known to be a parallelogram; the proof establishes the triangles are congruent, uses CPCTC to get equal angles, and then turns those back into parallel sides with the converse of the alternate interior angle theorem — a step students routinely try to use in the wrong direction.

How this one works

Prove the property

Prove∠S ≅ ∠U
RSTU is a parallelogram—Given
RS // UT, RU // ST—Definition of Parallelogram
∠SRT ≅ ∠UTR, ∠STR ≅ ∠URT—Alternate Interior Angles
RT ≅ RT—Reflexive Property
△RST ≅ △TUR—ASA
∠S ≅ ∠U—CPCTC
  1. Every proof of a parallelogram property runs the same way: draw a diagonal to make two triangles, use the parallel sides to find equal angles, notice the diagonal is shared, close with a congruence criterion, then read off the part with CPCTC. Here: given RSTU is a parallelogram, prove ∠S ≅ ∠U.
  2. 1. RSTU is a parallelogram. This fact is handed to you in the Given — nothing has to justify it.
  3. 2. RS // UT, RU // ST. A parallelogram is DEFINED as a quadrilateral with both pairs of opposite sides parallel — so naming it parallelogram is the same as saying those sides are parallel.
  4. 3. ∠SRT ≅ ∠UTR, ∠STR ≅ ∠URT. The two lines are parallel, so a transversal makes alternate interior angles congruent.
  5. 4. RT ≅ RT. A segment (or angle) is congruent to itself. The two triangles share this side, so it is equal in both by the reflexive property.
  6. 5. △RST ≅ △TUR. Two pairs of angles and the side between them are equal: ASA.
  7. 6. ∠S ≅ ∠U. Corresponding parts of congruent triangles are congruent — once the two triangles are congruent, every matching pair of sides and angles is equal, including pairs nobody gave you.

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