Worksheets · Geometry

Two-column proofs worksheet

A two-column proof is an argument written twice: what you know on the left, why you know it on the right. Every line comes from the Given, from a definition, from a property like the reflexive property, or from a theorem you have already proved. Nothing comes from how the picture looks. Once two triangles are congruent, CPCTC lets you carry any matching pair of sides or angles across, which is how most proofs finish.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Two-Column 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

    KLMNQRSGiven: LQ‾∥MR‾  ∠QLM and ∠MNS are supplementaryProve: MR‾∥NS‾StatementsReasons1.  LQ‾∥MR‾Given2.  ∠QLM and ∠LMR are supplementarySame-side interior angles of parallel lines are supplementary3.  ∠QLM and ∠MNS are supplementary  4.  ∠LMR≅∠MNS  5.  MR‾∥NS‾  
  2. PQRSTGiven: PS‾⊥QR‾ at T  ST‾ bisects ∠QSRProve: △STQ≅△STRStatementsReasons1.  PS‾⊥QR‾ at TGiven2.  ∠STQ and ∠STR are right anglesDefinition of perpendicular3.  ∠STQ≅∠STR  4.  ST‾≅ST‾Reflexive property5.  ST‾ bisects ∠QSRGiven6.  ∠QST≅∠RST  7.  △STQ≅△STR  
  3. 123Given: ∠1 and ∠2 are complementary  m∠2=60∘Prove: m∠3=30∘StatementsReasons1.  ∠1 and ∠2 are complementaryGiven2.  m∠2=60∘Given3.  m∠1+m∠2=90∘Definition of complementary angles4.  m∠1+60∘=90∘  5.  m∠1=30∘Subtraction property6.  ∠1≅∠3  7.  m∠1=m∠3Definition of congruent angles8.  m∠3=30∘  

  1. Fill in every reason

    RSTUGiven: RSTU is an isosceles trapezoid with RS‾∥UT‾Prove: RT‾≅SU‾Use the word bank.StatementsReasons1.  RSTU is an isosceles trapezoid with RS‾∥UT‾  2.  RU‾≅ST‾  3.  ∠RUT≅∠STU  4.  UT‾≅UT‾  5.  △RUT≅△STU  6.  RT‾≅SU‾  Word bank: SAS · CPCTC · Vertical angles are congruent · All radii of a circle are congruent · Opposite sides of a parallelogram are congruent · Base angles of an isosceles trapezoid are congruent · Given · Reflexive property · Definition of an isosceles trapezoid
  2. XVUWTGiven: ∠T≅∠W  TV‾≅WX‾  WV‾∥TU‾Prove: XV‾≅VU‾Use the word bank.StatementsReasons1.  ∠T≅∠W  2.  WV‾∥TU‾  3.  ∠WVX≅∠TUV  4.  TV‾≅WX‾  5.  △WXV≅△TVU  6.  XV‾≅VU‾  Word bank: Corresponding angles of parallel lines are congruent · Definition of a median · Distributive property · CPCTC · A parallelogram with one right angle is a rectangle · Definition of a rhombus · AAS · The segment from the center to a chord's midpoint is perpendicular to it · Given
  3. GHJKLGiven: GHJK is a parallelogram  GK‾≅JL‾  ∠JKL≅∠JLKProve: GHJK is a rhombusUse the word bank.StatementsReasons1.  GHJK is a parallelogram  2.  ∠JKL≅∠JLK  3.  JK‾≅JL‾  4.  GK‾≅JL‾  5.  GK‾≅JK‾  6.  GHJK is a rhombus  Word bank: Sides opposite congruent angles are congruent · Definition of a linear pair · Given · A parallelogram with two consecutive sides congruent is a rhombus · Transitive property · Definition of perpendicular · If two congruent angles form a linear pair, the lines are perpendicular · Addition property · Definition of an isosceles triangle

  1. Complete each proof, giving a reason for each step

    Given: 3x+2=x+32Prove: x=15StatementsReasons1.  3x+2=x+32Given2.  x=00  3.  x=00  4.  x=00  
  2. Given: x−33=1Prove: x=6StatementsReasons1.  x−33=1Given2.  x=00  3.  x=00  

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

Sigma Prep · sigmaprep.io/worksheets

Name

Two-Column 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

    KLMNQRSGiven: LQ‾∥MR‾  ∠QLM and ∠MNS are supplementaryProve: MR‾∥NS‾StatementsReasons1.  LQ‾∥MR‾Given2.  ∠QLM and ∠LMR are supplementarySame-side interior angles of parallel lines are supplementary3.  ∠QLM and ∠MNS are supplementary  4.  ∠LMR≅∠MNS  5.  MR‾∥NS‾  3.  Given4.  Supplements of the same angle are congruent5.  If corresponding angles are congruent, the lines are parallel
  2. PQRSTGiven: PS‾⊥QR‾ at T  ST‾ bisects ∠QSRProve: △STQ≅△STRStatementsReasons1.  PS‾⊥QR‾ at TGiven2.  ∠STQ and ∠STR are right anglesDefinition of perpendicular3.  ∠STQ≅∠STR  4.  ST‾≅ST‾Reflexive property5.  ST‾ bisects ∠QSRGiven6.  ∠QST≅∠RST  7.  △STQ≅△STR  3.  All right angles are congruent6.  Definition of an angle bisector7.  ASA
  3. 123Given: ∠1 and ∠2 are complementary  m∠2=60∘Prove: m∠3=30∘StatementsReasons1.  ∠1 and ∠2 are complementaryGiven2.  m∠2=60∘Given3.  m∠1+m∠2=90∘Definition of complementary angles4.  m∠1+60∘=90∘  5.  m∠1=30∘Subtraction property6.  ∠1≅∠3  7.  m∠1=m∠3Definition of congruent angles8.  m∠3=30∘  4.  Substitution6.  Vertical angles are congruent8.  Substitution

  1. Fill in every reason

    RSTUGiven: RSTU is an isosceles trapezoid with RS‾∥UT‾Prove: RT‾≅SU‾Use the word bank.StatementsReasons1.  RSTU is an isosceles trapezoid with RS‾∥UT‾  2.  RU‾≅ST‾  3.  ∠RUT≅∠STU  4.  UT‾≅UT‾  5.  △RUT≅△STU  6.  RT‾≅SU‾  Word bank: SAS · CPCTC · Vertical angles are congruent · All radii of a circle are congruent · Opposite sides of a parallelogram are congruent · Base angles of an isosceles trapezoid are congruent · Given · Reflexive property · Definition of an isosceles trapezoid1.  Given2.  Definition of an isosceles trapezoid3.  Base angles of an isosceles trapezoid are congruent4.  Reflexive property5.  SAS6.  CPCTC
  2. XVUWTGiven: ∠T≅∠W  TV‾≅WX‾  WV‾∥TU‾Prove: XV‾≅VU‾Use the word bank.StatementsReasons1.  ∠T≅∠W  2.  WV‾∥TU‾  3.  ∠WVX≅∠TUV  4.  TV‾≅WX‾  5.  △WXV≅△TVU  6.  XV‾≅VU‾  Word bank: Corresponding angles of parallel lines are congruent · Definition of a median · Distributive property · CPCTC · A parallelogram with one right angle is a rectangle · Definition of a rhombus · AAS · The segment from the center to a chord's midpoint is perpendicular to it · Given1.  Given2.  Given3.  Corresponding angles of parallel lines are congruent4.  Given5.  AAS6.  CPCTC
  3. GHJKLGiven: GHJK is a parallelogram  GK‾≅JL‾  ∠JKL≅∠JLKProve: GHJK is a rhombusUse the word bank.StatementsReasons1.  GHJK is a parallelogram  2.  ∠JKL≅∠JLK  3.  JK‾≅JL‾  4.  GK‾≅JL‾  5.  GK‾≅JK‾  6.  GHJK is a rhombus  Word bank: Sides opposite congruent angles are congruent · Definition of a linear pair · Given · A parallelogram with two consecutive sides congruent is a rhombus · Transitive property · Definition of perpendicular · If two congruent angles form a linear pair, the lines are perpendicular · Addition property · Definition of an isosceles triangle1.  Given2.  Given3.  Sides opposite congruent angles are congruent4.  Given5.  Transitive property6.  A parallelogram with two consecutive sides congruent is a rhombus

  1. Complete each proof, giving a reason for each step

    Given: 3x+2=x+32Prove: x=15StatementsReasons1.  3x+2=x+32Given2.  x=00  3.  x=00  4.  x=00  StatementsReasons1.  3x+2=x+32Given2.  2x+2=32Subtraction Property of Equality3.  2x=30Subtraction Property of Equality4.  x=15Division Property of Equality
  2. Given: x−33=1Prove: x=6StatementsReasons1.  x−33=1Given2.  x=00  3.  x=00  StatementsReasons1.  x−33=1Given2.  x−3=3Multiplication Property of Equality3.  x=6Addition Property of Equality

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

How to do these

Given AB‾≅CD‾, prove AC‾≅BD‾

  1. Line 1 is the Given, always.
  2. BC is congruent to itself by the reflexive property.
  3. Segment addition turns AB + BC into AC, and BC + CD into BD.
  4. Adding the same segment to congruent segments keeps them congruent.

The answer is Addition property.

Where students go wrong

Using something the picture shows but the Given never says. If two segments look equal and no mark or Given says so, you cannot use it.

Also called geometry proofs, statements and reasons or fill in the blank proofs.

What you can put on this worksheet

Fill in the missing reasons
MNPQRS
Given: QN‾≅RP‾
  ∠SQN≅∠SRP
Prove: △QNS≅△RPS
StatementsReasons1.  QN‾≅RP‾Given2.  ∠SQN≅∠SRPGiven3.  ∠NSQ≅∠PSR  4.  △QNS≅△RPS  
→ Answer
3.  Vertical angles are congruent
4.  AAS
Fill in every reason
MNPQRS
Given: QN‾≅RP‾
  ∠SQN≅∠SRP
Prove: △QNS≅△RPS
Use the word bank.
StatementsReasons1.  QN‾≅RP‾  2.  ∠SQN≅∠SRP  3.  ∠NSQ≅∠PSR  4.  △QNS≅△RPS  
Word bank: Given · Diagonals of a rhombus are perpendicular · Vertical angles are congruent · Definition of a segment bisector · In a circle, congruent central angles have congruent arcs · AAS · Tangent segments from the same external point are congruent
→ Answer
1.  Given
2.  Given
3.  Vertical angles are congruent
4.  AAS
Fill in the missing statements
MNPQRSTU
Given: MN‾∥PQ‾
  ∠MSR≅∠PQT
Prove: RS‾∥UQ‾
StatementsReasons1.  MN‾∥PQ‾Given2.    Corresponding angles of parallel lines are congruent3.    Given4.  ∠MSR≅∠MTUTransitive property5.  RS‾∥UQ‾If corresponding angles are congruent, the lines are parallel
→ Answer
2.  ∠MTU≅∠PQT
3.  ∠MSR≅∠PQT
Write the whole proof
MNPQRS
Given: QN‾≅RP‾
  ∠SQN≅∠SRP
Prove: △QNS≅△RPS
→ Answer
One possible proof:
StatementsReasons1.  QN‾≅RP‾Given2.  ∠SQN≅∠SRPGiven3.  ∠NSQ≅∠PSRVertical angles are congruent4.  △QNS≅△RPSAAS
Name the postulate that finishes the proof
KLNP
Given: ∠KNP≅∠KLP
  ∠LPK≅∠NPK
Prove: △PLK≅△PNK
Which congruence postulate or theorem proves the triangles congruent?
StatementsReasons1.  ∠KNP≅∠KLPGiven2.  ∠LPK≅∠NPKGiven3.  PK‾≅PK‾Reflexive property4.  △PLK≅△PNK  
→ Answer
AAS
Algebraic proofs
Given: 6−5x=−9Prove: x=3
StatementsReasons1.  6−5x=−9Given2.  x=00  3.  x=00  
→ Answer
StatementsReasons1.  6−5x=−9Given2.  −5x=−15Subtraction Property of Equality3.  x=3Division Property of Equality

Questions about these worksheets

Yes. Every two-column 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 two-column 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 6: fill in the missing reasons, fill in every reason, fill in the missing statements, write the whole proof, name the postulate that finishes the proof and algebraic proofs. Tick as many as you want and set how many of each, or let it spread them evenly.

Using something the picture shows but the Given never says. If two segments look equal and no mark or Given says so, you cannot use it.