Worksheets · Geometry

Proving lines parallel worksheet

These proofs run the parallel-line theorems backwards. Instead of being told the lines are parallel and concluding the angles are congruent, you are given the angles and must conclude the lines are parallel. Each theorem has a converse: congruent alternate interior angles prove the lines parallel, so do congruent corresponding angles, and so do supplementary same-side interior angles. Hard also goes the long way round: prove two triangles congruent, get the angles by CPCTC, then use the converse.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Proving Lines Parallel

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

    ABCDEGiven: AB‾∥CD‾  ∠ABC and ∠CDE are supplementaryProve: BC‾∥DE‾StatementsReasons1.  AB‾∥CD‾Given2.  ∠ABC≅∠BCDAlternate interior angles of parallel lines are congruent3.  m∠ABC=m∠BCD  4.  ∠ABC and ∠CDE are supplementaryGiven5.  m∠ABC+m∠CDE=180∘Definition of supplementary angles6.  m∠BCD+m∠CDE=180∘  7.  ∠BCD and ∠CDE are supplementaryDefinition of supplementary angles8.  BC‾∥DE‾  
  2. STUVGiven: ∠STU and ∠UVS are right angles  ST‾≅UV‾Prove: SV‾∥TU‾StatementsReasons1.∠STU and ∠UVS are right anglesGiven2.ST‾≅UV‾3.SU‾≅US‾4.△STU≅△UVSHL5.∠TUS≅∠VSU6.SV‾∥TU‾If alternate interior angles are congruent, the lines are parallel
  3. WXYZTGiven: Y is the midpoint of XT‾  WX‾≅ZY‾  WY‾≅ZT‾Prove: WX‾∥ZY‾StatementsReasons1.  Y is the midpoint of XT‾Given2.  XY‾≅YT‾  3.  WX‾≅ZY‾  4.  WY‾≅ZT‾Given5.  △WXY≅△ZYTSSS6.  ∠WXY≅∠ZYTCPCTC7.  WX‾∥ZY‾  
  4. KLMNPQGiven: KL‾∥NP‾  m∠LKM=m∠PNQProve: KM‾∥NQ‾StatementsReasons1.  KL‾∥NP‾Given2.  ∠LKN≅∠KNPAlternate interior angles of parallel lines are congruent3.  m∠LKN=m∠KNPDefinition of congruent angles4.  m∠LKM=m∠PNQ  5.  m∠LKN+m∠NKM=m∠LKMAngle addition postulate6.  m∠PNK+m∠KNQ=m∠PNQAngle addition postulate7.  m∠NKM=m∠KNQSubtraction property8.  ∠NKM≅∠KNQ  9.  KM‾∥NQ‾  

  1. Fill in every reason

    TUVWXGiven: UW‾ bisects ∠TUV  XU‾≅XW‾Prove: WX‾∥UV‾Use the word bank.StatementsReasons1.  UW‾ bisects ∠TUV  2.  ∠XUW≅∠WUV  3.  XU‾≅XW‾  4.  ∠XUW≅∠XWU  5.  ∠XWU≅∠WUV  6.  WX‾∥UV‾  Word bank: ASA · If alternate interior angles are congruent, the lines are parallel · Subtraction property · Transitive property · CPCTC · Definition of complementary angles · Given · Definition of an angle bisector · Base angles of an isosceles triangle are congruent
  2. PQRSTGiven: PT‾≅ST‾  QT‾≅RT‾Prove: PQ‾∥RS‾Use the word bank.StatementsReasons1.  PT‾≅ST‾  2.  QT‾≅RT‾  3.  ∠PTQ≅∠STR  4.  △PQT≅△SRT  5.  ∠QPT≅∠RST  6.  PQ‾∥RS‾  Word bank: Given · Vertical angles are congruent · Sides opposite congruent angles are congruent · Third angles theorem · Definition of perpendicular · Exterior angle theorem · If alternate interior angles are congruent, the lines are parallel · SAS · CPCTC
  3. 12345678ℓmtGiven: ∠1 and ∠7 are supplementaryProve: ℓ∥mUse the word bank.StatementsReasons1.  ∠1 and ∠7 are supplementary  2.  ∠1≅∠4  3.  ∠7≅∠6  4.  m∠1+m∠7=180∘  5.  m∠1=m∠4 and m∠7=m∠6  6.  m∠4+m∠6=180∘  7.  ∠4 and ∠6 are supplementary  8.  ℓ∥m  Word bank: Definition of a right angle · If same-side interior angles are supplementary, the lines are parallel · Substitution · Definition of congruent angles · CPCTC · Vertical angles are congruent · Given · Angle addition postulate · If alternate interior angles are congruent, the lines are parallel · All right angles are congruent · Definition of supplementary angles
  4. FGHIJLGiven: FJ→ meets HI‾ at L  m∠GFJ+m∠IHJ=m∠FJHProve: FG‾∥HI‾Use the word bank.StatementsReasons1.  FJ→ meets HI‾ at L  2.  m∠GFJ+m∠IHJ=m∠FJH  3.  m∠JLH+m∠IHJ=m∠FJH  4.  m∠GFJ+m∠IHJ=m∠JLH+m∠IHJ  5.  m∠GFJ=m∠JLH  6.  ∠GFJ≅∠JLH  7.  FG‾∥HI‾  Word bank: Given · Substitution · Definition of congruent angles · If alternate interior angles are congruent, the lines are parallel · Definition of supplementary angles · AAS · Supplements of congruent angles are congruent · Exterior angle theorem · Subtraction property · Definition of complementary angles

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

Sigma Prep · sigmaprep.io/worksheets

Name

Proving Lines ParallelAnswer 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

    ABCDEGiven: AB‾∥CD‾  ∠ABC and ∠CDE are supplementaryProve: BC‾∥DE‾StatementsReasons1.  AB‾∥CD‾Given2.  ∠ABC≅∠BCDAlternate interior angles of parallel lines are congruent3.  m∠ABC=m∠BCD  4.  ∠ABC and ∠CDE are supplementaryGiven5.  m∠ABC+m∠CDE=180∘Definition of supplementary angles6.  m∠BCD+m∠CDE=180∘  7.  ∠BCD and ∠CDE are supplementaryDefinition of supplementary angles8.  BC‾∥DE‾  3.  Definition of congruent angles6.  Substitution8.  If same-side interior angles are supplementary, the lines are parallel
  2. STUVGiven: ∠STU and ∠UVS are right angles  ST‾≅UV‾Prove: SV‾∥TU‾StatementsReasons1.∠STU and ∠UVS are right anglesGiven2.ST‾≅UV‾3.SU‾≅US‾4.△STU≅△UVSHL5.∠TUS≅∠VSU6.SV‾∥TU‾If alternate interior angles are congruent, the lines are parallel2.  Given3.  Reflexive property5.  CPCTC
  3. WXYZTGiven: Y is the midpoint of XT‾  WX‾≅ZY‾  WY‾≅ZT‾Prove: WX‾∥ZY‾StatementsReasons1.  Y is the midpoint of XT‾Given2.  XY‾≅YT‾  3.  WX‾≅ZY‾  4.  WY‾≅ZT‾Given5.  △WXY≅△ZYTSSS6.  ∠WXY≅∠ZYTCPCTC7.  WX‾∥ZY‾  2.  Definition of midpoint3.  Given7.  If corresponding angles are congruent, the lines are parallel
  4. KLMNPQGiven: KL‾∥NP‾  m∠LKM=m∠PNQProve: KM‾∥NQ‾StatementsReasons1.  KL‾∥NP‾Given2.  ∠LKN≅∠KNPAlternate interior angles of parallel lines are congruent3.  m∠LKN=m∠KNPDefinition of congruent angles4.  m∠LKM=m∠PNQ  5.  m∠LKN+m∠NKM=m∠LKMAngle addition postulate6.  m∠PNK+m∠KNQ=m∠PNQAngle addition postulate7.  m∠NKM=m∠KNQSubtraction property8.  ∠NKM≅∠KNQ  9.  KM‾∥NQ‾  4.  Given8.  Definition of congruent angles9.  If alternate interior angles are congruent, the lines are parallel

  1. Fill in every reason

    TUVWXGiven: UW‾ bisects ∠TUV  XU‾≅XW‾Prove: WX‾∥UV‾Use the word bank.StatementsReasons1.  UW‾ bisects ∠TUV  2.  ∠XUW≅∠WUV  3.  XU‾≅XW‾  4.  ∠XUW≅∠XWU  5.  ∠XWU≅∠WUV  6.  WX‾∥UV‾  Word bank: ASA · If alternate interior angles are congruent, the lines are parallel · Subtraction property · Transitive property · CPCTC · Definition of complementary angles · Given · Definition of an angle bisector · Base angles of an isosceles triangle are congruent1.  Given2.  Definition of an angle bisector3.  Given4.  Base angles of an isosceles triangle are congruent5.  Transitive property6.  If alternate interior angles are congruent, the lines are parallel
  2. PQRSTGiven: PT‾≅ST‾  QT‾≅RT‾Prove: PQ‾∥RS‾Use the word bank.StatementsReasons1.  PT‾≅ST‾  2.  QT‾≅RT‾  3.  ∠PTQ≅∠STR  4.  △PQT≅△SRT  5.  ∠QPT≅∠RST  6.  PQ‾∥RS‾  Word bank: Given · Vertical angles are congruent · Sides opposite congruent angles are congruent · Third angles theorem · Definition of perpendicular · Exterior angle theorem · If alternate interior angles are congruent, the lines are parallel · SAS · CPCTC1.  Given2.  Given3.  Vertical angles are congruent4.  SAS5.  CPCTC6.  If alternate interior angles are congruent, the lines are parallel
  3. 12345678ℓmtGiven: ∠1 and ∠7 are supplementaryProve: ℓ∥mUse the word bank.StatementsReasons1.  ∠1 and ∠7 are supplementary  2.  ∠1≅∠4  3.  ∠7≅∠6  4.  m∠1+m∠7=180∘  5.  m∠1=m∠4 and m∠7=m∠6  6.  m∠4+m∠6=180∘  7.  ∠4 and ∠6 are supplementary  8.  ℓ∥m  Word bank: Definition of a right angle · If same-side interior angles are supplementary, the lines are parallel · Substitution · Definition of congruent angles · CPCTC · Vertical angles are congruent · Given · Angle addition postulate · If alternate interior angles are congruent, the lines are parallel · All right angles are congruent · Definition of supplementary angles1.  Given2.  Vertical angles are congruent3.  Vertical angles are congruent4.  Definition of supplementary angles5.  Definition of congruent angles6.  Substitution7.  Definition of supplementary angles8.  If same-side interior angles are supplementary, the lines are parallel
  4. FGHIJLGiven: FJ→ meets HI‾ at L  m∠GFJ+m∠IHJ=m∠FJHProve: FG‾∥HI‾Use the word bank.StatementsReasons1.  FJ→ meets HI‾ at L  2.  m∠GFJ+m∠IHJ=m∠FJH  3.  m∠JLH+m∠IHJ=m∠FJH  4.  m∠GFJ+m∠IHJ=m∠JLH+m∠IHJ  5.  m∠GFJ=m∠JLH  6.  ∠GFJ≅∠JLH  7.  FG‾∥HI‾  Word bank: Given · Substitution · Definition of congruent angles · If alternate interior angles are congruent, the lines are parallel · Definition of supplementary angles · AAS · Supplements of congruent angles are congruent · Exterior angle theorem · Subtraction property · Definition of complementary angles1.  Given2.  Given3.  Exterior angle theorem4.  Substitution5.  Subtraction property6.  Definition of congruent angles7.  If alternate interior angles are congruent, the lines are parallel

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

How to do these

Given ∠3≅∠6, why are the lines parallel?

  1. Angles 3 and 6 are alternate interior angles.
  2. They are congruent, so the converse applies.

The answer is If alternate interior angles are congruent, the lines are parallel.

Where students go wrong

Quoting the forward theorem when you need the converse. "Alternate interior angles are congruent" assumes the lines are parallel, which is the thing you are trying to prove.

Also called converse of alternate interior angles, proving parallel lines, parallel line proofs or transversal proofs.

What you can put on this worksheet

Fill in the missing reasons
KLMNPQRS
Given: KL‾∥MN‾
  ∠KQP≅∠MNR
Prove: PQ‾∥SN‾
StatementsReasons1.  KL‾∥MN‾Given2.  ∠KRS≅∠MNR  3.  ∠KQP≅∠MNR  4.  ∠KQP≅∠KRSTransitive property5.  PQ‾∥SN‾If corresponding angles are congruent, the lines are parallel
→ Answer
2.  Corresponding angles of parallel lines are congruent
3.  Given
Fill in every reason
KLMNPQRS
Given: KL‾∥MN‾
  ∠KQP≅∠MNR
Prove: PQ‾∥SN‾
Use the word bank.
StatementsReasons1.  KL‾∥MN‾  2.  ∠KRS≅∠MNR  3.  ∠KQP≅∠MNR  4.  ∠KQP≅∠KRS  5.  PQ‾∥SN‾  
Word bank: Segment addition postulate · Corresponding angles of parallel lines are congruent · Given · If corresponding angles are congruent, the lines are parallel · If same-side interior angles are supplementary, the lines are parallel · CPCTC · Transitive property · All right angles are congruent
→ Answer
1.  Given
2.  Corresponding angles of parallel lines are congruent
3.  Given
4.  Transitive property
5.  If corresponding angles are congruent, the lines are parallel
Fill in the missing statements
KLMNP
Given: KL‾≅KM‾
  ∠KNP≅∠KML
Prove: NP‾∥LM‾
StatementsReasons1.  KL‾≅KM‾Given2.  ∠KLM≅∠KMLBase angles of an isosceles triangle are congruent3.    Given4.    Transitive property5.  NP‾∥LM‾If corresponding angles are congruent, the lines are parallel
→ Answer
3.  ∠KNP≅∠KML
4.  ∠KNP≅∠KLM
Write the whole proof
KLMNPQRS
Given: KL‾∥MN‾
  ∠KQP≅∠MNR
Prove: PQ‾∥SN‾
→ Answer
One possible proof:
StatementsReasons1.  KL‾∥MN‾Given2.  ∠KRS≅∠MNRCorresponding angles of parallel lines are congruent3.  ∠KQP≅∠MNRGiven4.  ∠KQP≅∠KRSTransitive property5.  PQ‾∥SN‾If corresponding angles are congruent, the lines are parallel
Solve for x from the drawing
Find x. — two parallel lines cut by a transversal, with angles marked → x=4
The x that makes the lines parallel
Two alternate interior angles measure (5x+4)∘ and (6x−1)∘. Find x so that ℓ∥m. → x=5

Questions about these worksheets

Yes. Every proving lines parallel 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 proving lines parallel 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, solve for x from the drawing and the x that makes the lines parallel. Tick as many as you want and set how many of each, or let it spread them evenly.

Quoting the forward theorem when you need the converse. "Alternate interior angles are congruent" assumes the lines are parallel, which is the thing you are trying to prove.