Worksheets · Geometry

Parallelogram proofs worksheet

Draw a diagonal and a parallelogram becomes two triangles, which is how every one of these proofs works. The definition gives you both pairs of opposite sides parallel, parallel lines give congruent alternate interior angles, the diagonal is shared so it is congruent to itself, and ASA finishes it. CPCTC then hands you whichever pair of opposite sides or angles you were asked for.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Parallelogram Proofs

Date Period

Complete each proof. From a figure you may assume points on a line, points between others, and where lines meet; do not assume congruent parts, right angles or parallel lines.

  1. Fill in the missing reasons

    KLMNPQRGiven: KLMN is a parallelogram  PQRN is a parallelogramProve: ∠L≅∠QStatementsReasons1.  KLMN is a parallelogramGiven2.  PQRN is a parallelogram  3.  ∠L≅∠KNMOpposite angles of a parallelogram are congruent4.  ∠KNM≅∠PNR  5.  ∠PNR≅∠QOpposite angles of a parallelogram are congruent6.  ∠L≅∠Q  
  2. WXYZTGiven: T is the midpoint of WY‾ and XZ‾Prove: WX‾∥YZ‾StatementsReasons1.  T is the midpoint of WY‾ and XZ‾Given2.  WT‾≅YT‾  3.  XT‾≅ZT‾  4.  ∠WTX≅∠YTZVertical angles are congruent5.  △WTX≅△YTZ  6.  ∠XWT≅∠ZYTCPCTC7.  WX‾∥YZ‾If alternate interior angles are congruent, the lines are parallel
  3. MNPQGiven: MNPQ is a rhombusProve: MP‾ bisects ∠NMQStatementsReasons1.  MNPQ is a rhombusGiven2.  MNPQ is a parallelogram  3.  MN‾∥QP‾Definition of a parallelogram4.  ∠NMP≅∠QPMAlternate interior angles of parallel lines are congruent5.  QM‾≅QP‾Definition of a rhombus6.  ∠QMP≅∠QPM  7.  ∠NMP≅∠QMP  8.  MP‾ bisects ∠NMQDefinition of an angle bisector
  4. STUVGiven: STUV is a parallelogramProve: ∠S≅∠UStatementsReasons1.  STUV is a parallelogramGiven2.  ST‾∥VU‾  3.  ∠S and ∠V are supplementary  4.  TU‾∥SV‾  5.  ∠U and ∠V are supplementarySame-side interior angles of parallel lines are supplementary6.  ∠S≅∠USupplements of congruent angles are congruent

  1. Fill in every reason

    RSTUWGiven: RSTU is a parallelogram  RU‾≅TW‾  ∠TUW≅∠TWUProve: RSTU is a rhombusUse the word bank.StatementsReasons1.  RSTU is a parallelogram  2.  ∠TUW≅∠TWU  3.  TU‾≅TW‾  4.  RU‾≅TW‾  5.  RU‾≅TU‾  6.  RSTU is a rhombus  Word bank: Given · Alternate interior angles of parallel lines are congruent · Transitive property · Consecutive angles of a parallelogram are supplementary · A parallelogram with two consecutive sides congruent is a rhombus · Sides opposite congruent angles are congruent · All right angles are congruent · If corresponding angles are congruent, the lines are parallel · If two angles are congruent and supplementary, each is a right angle
  2. FGHIGiven: FG‾∥IH‾  ∠F≅∠HProve: FGHI is a parallelogramUse the word bank.StatementsReasons1.  FG‾∥IH‾  2.  ∠F and ∠I are supplementary  3.  ∠F≅∠H  4.  m∠F=m∠H  5.  m∠F+m∠I=180∘  6.  m∠H+m∠I=180∘  7.  ∠H and ∠I are supplementary  8.  FI‾∥GH‾  9.  FGHI is a parallelogram  Word bank: Definition of a parallelogram · Definition of midpoint · If both pairs of opposite sides are parallel, the figure is a parallelogram · If same-side interior angles are supplementary, the lines are parallel · Linear pairs are supplementary · Halves of congruent angles are congruent · If alternate interior angles are congruent, the lines are parallel · Definition of congruent angles · Same-side interior angles of parallel lines are supplementary · Substitution · Given · Definition of supplementary angles · Definition of perpendicular
  3. ABCDEGiven: ABCD is a rectangle  E is the midpoint of AB‾Prove: DE‾≅CE‾Use the word bank.StatementsReasons1.  ABCD is a rectangle  2.  ABCD is a parallelogram  3.  AD‾≅BC‾  4.  ∠A and ∠B are right angles  5.  ∠A≅∠B  6.  E is the midpoint of AB‾  7.  AE‾≅BE‾  8.  △DAE≅△CBE  9.  DE‾≅CE‾  Word bank: Definition of midpoint · Definition of a rectangle · Opposite sides of a parallelogram are congruent · Definition of a rhombus · All right angles are congruent · Same-side interior angles of parallel lines are supplementary · CPCTC · Definition of supplementary angles · SAS · Given · The acute angles of a right triangle are complementary · AAS
  4. GHJKLGiven: GJ‾ and HK‾ meet at L  GHJK is a parallelogramProve: HL‾≅KL‾Use the word bank.StatementsReasons1.  GJ‾ and HK‾ meet at L  2.  GHJK is a parallelogram  3.  GH‾∥JK‾  4.  ∠GHL≅∠JKL  5.  ∠GLH≅∠JLK  6.  GH‾≅JK‾  7.  △GHL≅△JKL  8.  HL‾≅KL‾  Word bank: Segment addition postulate · Definition of a parallelogram · One pair of opposite sides parallel and congruent makes a parallelogram · Vertical angles are congruent · Opposite sides of a parallelogram are congruent · Given · Definition of a rhombus · AAS · Definition of complementary angles · Alternate interior angles of parallel lines are congruent · CPCTC

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Parallelogram ProofsAnswer keyVersion 1

Date Period

Complete each proof. From a figure you may assume points on a line, points between others, and where lines meet; do not assume congruent parts, right angles or parallel lines.

  1. Fill in the missing reasons

    KLMNPQRGiven: KLMN is a parallelogram  PQRN is a parallelogramProve: ∠L≅∠QStatementsReasons1.  KLMN is a parallelogramGiven2.  PQRN is a parallelogram  3.  ∠L≅∠KNMOpposite angles of a parallelogram are congruent4.  ∠KNM≅∠PNR  5.  ∠PNR≅∠QOpposite angles of a parallelogram are congruent6.  ∠L≅∠Q  2.  Given4.  Vertical angles are congruent6.  Transitive property
  2. WXYZTGiven: T is the midpoint of WY‾ and XZ‾Prove: WX‾∥YZ‾StatementsReasons1.  T is the midpoint of WY‾ and XZ‾Given2.  WT‾≅YT‾  3.  XT‾≅ZT‾  4.  ∠WTX≅∠YTZVertical angles are congruent5.  △WTX≅△YTZ  6.  ∠XWT≅∠ZYTCPCTC7.  WX‾∥YZ‾If alternate interior angles are congruent, the lines are parallel2.  Definition of midpoint3.  Definition of midpoint5.  SAS
  3. MNPQGiven: MNPQ is a rhombusProve: MP‾ bisects ∠NMQStatementsReasons1.  MNPQ is a rhombusGiven2.  MNPQ is a parallelogram  3.  MN‾∥QP‾Definition of a parallelogram4.  ∠NMP≅∠QPMAlternate interior angles of parallel lines are congruent5.  QM‾≅QP‾Definition of a rhombus6.  ∠QMP≅∠QPM  7.  ∠NMP≅∠QMP  8.  MP‾ bisects ∠NMQDefinition of an angle bisector2.  Definition of a rhombus6.  Base angles of an isosceles triangle are congruent7.  Transitive property
  4. STUVGiven: STUV is a parallelogramProve: ∠S≅∠UStatementsReasons1.  STUV is a parallelogramGiven2.  ST‾∥VU‾  3.  ∠S and ∠V are supplementary  4.  TU‾∥SV‾  5.  ∠U and ∠V are supplementarySame-side interior angles of parallel lines are supplementary6.  ∠S≅∠USupplements of congruent angles are congruent2.  Definition of a parallelogram3.  Same-side interior angles of parallel lines are supplementary4.  Definition of a parallelogram

  1. Fill in every reason

    RSTUWGiven: RSTU is a parallelogram  RU‾≅TW‾  ∠TUW≅∠TWUProve: RSTU is a rhombusUse the word bank.StatementsReasons1.  RSTU is a parallelogram  2.  ∠TUW≅∠TWU  3.  TU‾≅TW‾  4.  RU‾≅TW‾  5.  RU‾≅TU‾  6.  RSTU is a rhombus  Word bank: Given · Alternate interior angles of parallel lines are congruent · Transitive property · Consecutive angles of a parallelogram are supplementary · A parallelogram with two consecutive sides congruent is a rhombus · Sides opposite congruent angles are congruent · All right angles are congruent · If corresponding angles are congruent, the lines are parallel · If two angles are congruent and supplementary, each is a right angle1.  Given2.  Given3.  Sides opposite congruent angles are congruent4.  Given5.  Transitive property6.  A parallelogram with two consecutive sides congruent is a rhombus
  2. FGHIGiven: FG‾∥IH‾  ∠F≅∠HProve: FGHI is a parallelogramUse the word bank.StatementsReasons1.  FG‾∥IH‾  2.  ∠F and ∠I are supplementary  3.  ∠F≅∠H  4.  m∠F=m∠H  5.  m∠F+m∠I=180∘  6.  m∠H+m∠I=180∘  7.  ∠H and ∠I are supplementary  8.  FI‾∥GH‾  9.  FGHI is a parallelogram  Word bank: Definition of a parallelogram · Definition of midpoint · If both pairs of opposite sides are parallel, the figure is a parallelogram · If same-side interior angles are supplementary, the lines are parallel · Linear pairs are supplementary · Halves of congruent angles are congruent · If alternate interior angles are congruent, the lines are parallel · Definition of congruent angles · Same-side interior angles of parallel lines are supplementary · Substitution · Given · Definition of supplementary angles · Definition of perpendicular1.  Given2.  Same-side interior angles of parallel lines are supplementary3.  Given4.  Definition of congruent angles5.  Definition of supplementary angles6.  Substitution7.  Definition of supplementary angles8.  If same-side interior angles are supplementary, the lines are parallel9.  If both pairs of opposite sides are parallel, the figure is a parallelogramor: Definition of a parallelogram
  3. ABCDEGiven: ABCD is a rectangle  E is the midpoint of AB‾Prove: DE‾≅CE‾Use the word bank.StatementsReasons1.  ABCD is a rectangle  2.  ABCD is a parallelogram  3.  AD‾≅BC‾  4.  ∠A and ∠B are right angles  5.  ∠A≅∠B  6.  E is the midpoint of AB‾  7.  AE‾≅BE‾  8.  △DAE≅△CBE  9.  DE‾≅CE‾  Word bank: Definition of midpoint · Definition of a rectangle · Opposite sides of a parallelogram are congruent · Definition of a rhombus · All right angles are congruent · Same-side interior angles of parallel lines are supplementary · CPCTC · Definition of supplementary angles · SAS · Given · The acute angles of a right triangle are complementary · AAS1.  Given2.  Definition of a rectangle3.  Opposite sides of a parallelogram are congruent4.  Definition of a rectangle5.  All right angles are congruent6.  Given7.  Definition of midpoint8.  SAS9.  CPCTC
  4. GHJKLGiven: GJ‾ and HK‾ meet at L  GHJK is a parallelogramProve: HL‾≅KL‾Use the word bank.StatementsReasons1.  GJ‾ and HK‾ meet at L  2.  GHJK is a parallelogram  3.  GH‾∥JK‾  4.  ∠GHL≅∠JKL  5.  ∠GLH≅∠JLK  6.  GH‾≅JK‾  7.  △GHL≅△JKL  8.  HL‾≅KL‾  Word bank: Segment addition postulate · Definition of a parallelogram · One pair of opposite sides parallel and congruent makes a parallelogram · Vertical angles are congruent · Opposite sides of a parallelogram are congruent · Given · Definition of a rhombus · AAS · Definition of complementary angles · Alternate interior angles of parallel lines are congruent · CPCTC1.  Given2.  Given3.  Definition of a parallelogram4.  Alternate interior angles of parallel lines are congruent5.  Vertical angles are congruent6.  Opposite sides of a parallelogram are congruent7.  AAS8.  CPCTC

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Given ABCD is a parallelogram, why is AB‾≅CD‾?

  1. Draw the diagonal AC, shared by both triangles.
  2. Both pairs of opposite sides are parallel, giving two pairs of alternate interior angles.
  3. ASA proves the triangles congruent, then CPCTC.

The answer is CPCTC.

Where students go wrong

Assuming the property you are proving. If the question asks you to prove opposite sides congruent, you cannot start by saying they are congruent because it is a parallelogram.

Also called quadrilateral proofs, proving a quadrilateral is a parallelogram, opposite sides of a parallelogram or properties of parallelograms proof.

What you can put on this worksheet

Fill in the missing reasons
MNPQ
Given: NP‾≅QM‾
  NP‾∥MQ‾
Prove: △MNP≅△PQM
StatementsReasons1.  NP‾≅QM‾Given2.  NP‾∥MQ‾Given3.  ∠NPM≅∠QMP  4.  MP‾≅MP‾  5.  △MNP≅△PQMSAS
→ Answer
3.  Alternate interior angles of parallel lines are congruent
4.  Reflexive property
Fill in every reason
MNPQ
Given: NP‾≅QM‾
  NP‾∥MQ‾
Prove: △MNP≅△PQM
Use the word bank.
StatementsReasons1.  NP‾≅QM‾  2.  NP‾∥MQ‾  3.  ∠NPM≅∠QMP  4.  MP‾≅MP‾  5.  △MNP≅△PQM  
Word bank: Parts of parallel lines are parallel · Supplements of congruent angles are congruent · Alternate interior angles of parallel lines are congruent · Given · If two congruent angles form a linear pair, the lines are perpendicular · SAS · Halves of congruent angles are congruent · Reflexive property
→ Answer
1.  Given
2.  Given
3.  Alternate interior angles of parallel lines are congruent
4.  Reflexive property
5.  SAS
Fill in the missing statements
MNPQR
Given: MP‾ and NQ‾ meet at R
  MQ‾∥NP‾
  MQ‾≅NP‾
Prove: NR‾≅QR‾
StatementsReasons1.MP‾ and NQ‾ meet at RGiven2.Given3.Given4.MNPQ is a parallelogramOne pair of opposite sides parallel and congruent makes a parallelogram5.NR‾≅QR‾Diagonals of a parallelogram bisect each other
→ Answer
2.  MQ‾∥NP‾
3.  MQ‾≅NP‾
Write the whole proof
MNPQ
Given: NP‾≅QM‾
  NP‾∥MQ‾
Prove: △MNP≅△PQM
→ Answer
One possible proof:
StatementsReasons1.  NP‾≅QM‾Given2.  NP‾∥MQ‾Given3.  ∠NPM≅∠QMPAlternate interior angles of parallel lines are congruent4.  MP‾≅MP‾Reflexive property5.  △MNP≅△PQMSAS
Name the postulate that finishes the proof
KLMN
Given: LM‾≅NK‾
  LM‾∥KN‾
Prove: △KLM≅△MNK
Which congruence postulate or theorem proves the triangles congruent?
StatementsReasons1.  LM‾≅NK‾Given2.  LM‾∥KN‾Given3.  ∠LMK≅∠NKMAlternate interior angles of parallel lines are congruent4.  KM‾≅KM‾Reflexive property5.  △KLM≅△MNK  
→ Answer
SAS

Questions about these worksheets

Yes. Every parallelogram proofs sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Geometry is usually taken around 9th or 10th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the parallelogram proofs questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 5: fill in the missing reasons, fill in every reason, fill in the missing statements, write the whole proof and name the postulate that finishes the proof. Tick as many as you want and set how many of each, or let it spread them evenly.

Assuming the property you are proving. If the question asks you to prove opposite sides congruent, you cannot start by saying they are congruent because it is a parallelogram.