Worksheets · Geometry

CPCTC proofs worksheet

CPCTC stands for corresponding parts of congruent triangles are congruent, and it is always the last move, never the first. Prove the two triangles congruent by SSS, SAS, ASA, AAS or HL, and only then may you say that any other matching pair of sides or angles is congruent. Writing CPCTC before the triangles are congruent is the one thing these sheets are built to catch.

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

Sigma Prep · sigmaprep.io/worksheets

Name

CPCTC 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

    KLMNPQGiven: KL‾≅MN‾  ∠KMP≅∠NLQ  ∠P≅∠QProve: KP‾≅NQ‾StatementsReasons1.  KL‾≅MN‾Given2.  KL+LM=KM and LM+MN=LNSegment addition postulate3.  KL=MN  4.  KL+LM=MN+LMAddition property5.  KM=NLSubstitution6.  KM‾≅NL‾  7.  ∠KMP≅∠NLQGiven8.  ∠P≅∠QGiven9.  △KMP≅△NLQAAS10.  KP‾≅NQ‾  
  2. ABCDGiven: ABCD is a parallelogramProve: ∠A≅∠CStatementsReasons1.  ABCD is a parallelogramGiven2.  AB‾∥DC‾Definition of a parallelogram3.  ∠ABD≅∠CDB  4.  AD‾∥BC‾  5.  ∠ADB≅∠CBDAlternate interior angles of parallel lines are congruent6.  BD‾≅BD‾Reflexive property7.  △ABD≅△CDB  8.  ∠A≅∠CCPCTC
  3. DFGHEJGiven: DE‾≅GJ‾  DF‾≅GH‾  FE‾≅HJ‾Prove: GJ‾∥DE‾StatementsReasons1.  DE‾≅GJ‾Given2.  DF‾≅GH‾  3.  FE‾≅HJ‾Given4.  △DFE≅△GHJ  5.  ∠FDE≅∠HGJ  6.  GJ‾∥DE‾If corresponding angles are congruent, the lines are parallel
  4. JKLMGiven: JK‾≅LK‾  KM‾⊥JL‾Prove: JM‾≅LM‾StatementsReasons1.  JK‾≅LK‾Given2.  KM‾⊥JL‾Given3.  ∠KMJ and ∠KML are right angles  4.  KM‾≅KM‾Reflexive property5.  △JKM≅△LKM  6.  JM‾≅LM‾  

  1. Fill in every reason

    RSWTUGiven: W is the midpoint of RU‾ and ST‾Prove: RS‾∥TU‾Use the word bank.StatementsReasons1.  W is the midpoint of RU‾ and ST‾  2.  RW‾≅UW‾  3.  SW‾≅TW‾  4.  ∠RWS≅∠UWT  5.  △RSW≅△UTW  6.  ∠SRW≅∠TUW  7.  RS‾∥TU‾  Word bank: Given · Same-side interior angles of parallel lines are supplementary · Definition of a parallelogram · Corresponding angles of parallel lines are congruent · SAS · Definition of midpoint · Opposite angles of a parallelogram are congruent · CPCTC · If alternate interior angles are congruent, the lines are parallel · Vertical angles are congruent
  2. QPRSGiven: SP‾⊥QP‾  SR‾⊥QR‾  SP‾≅SR‾Prove: QS‾ bisects ∠PQRUse the word bank.StatementsReasons1.  SP‾⊥QP‾  2.  SR‾⊥QR‾  3.  ∠QPS and ∠QRS are right angles  4.  SP‾≅SR‾  5.  QS‾≅QS‾  6.  △QPS≅△QRS  7.  ∠PQS≅∠RQS  8.  QS‾ bisects ∠PQR  Word bank: Reflexive property · Opposite angles of a parallelogram are congruent · Definition of an angle bisector · Given · The acute angles of a right triangle are complementary · AAS · Linear pairs are supplementary · Definition of perpendicular · HL · Angle addition postulate · CPCTC
  3. FGHIJKGiven: IJ‾∥FG‾  J is the midpoint of HG‾  IJ‾≅KG‾Prove: KJ‾∥FH‾Use the word bank.StatementsReasons1.  J is the midpoint of HG‾  2.  HJ‾≅JG‾  3.  IJ‾∥FG‾  4.  ∠HJI≅∠JGK  5.  IJ‾≅KG‾  6.  △HJI≅△JGK  7.  ∠IHJ≅∠KJG  8.  KJ‾∥FH‾  Word bank: Definition of midpoint · Corresponding angles of parallel lines are congruent · If both pairs of opposite sides are congruent, it is a parallelogram · Segment addition postulate · Definition of perpendicular · CPCTC · Given · SAS · The acute angles of a right triangle are complementary · If corresponding angles are congruent, the lines are parallel · Base angles of an isosceles triangle are congruent
  4. TYXZWGiven: ∠W≅∠Z  WY‾≅ZT‾  ZY‾∥WX‾Prove: TY‾≅YX‾Use the word bank.StatementsReasons1.  ∠W≅∠Z  2.  ZY‾∥WX‾  3.  ∠ZYT≅∠WXY  4.  WY‾≅ZT‾  5.  △ZTY≅△WYX  6.  TY‾≅YX‾  Word bank: Corresponding angles of parallel lines are congruent · AAS · Definition of a parallelogram · If alternate interior angles are congruent, the lines are parallel · Complements of the same angle are congruent · ASA · CPCTC · Given · Definition of a rectangle

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

Sigma Prep · sigmaprep.io/worksheets

Name

CPCTC 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

    KLMNPQGiven: KL‾≅MN‾  ∠KMP≅∠NLQ  ∠P≅∠QProve: KP‾≅NQ‾StatementsReasons1.  KL‾≅MN‾Given2.  KL+LM=KM and LM+MN=LNSegment addition postulate3.  KL=MN  4.  KL+LM=MN+LMAddition property5.  KM=NLSubstitution6.  KM‾≅NL‾  7.  ∠KMP≅∠NLQGiven8.  ∠P≅∠QGiven9.  △KMP≅△NLQAAS10.  KP‾≅NQ‾  3.  Definition of congruent segments6.  Definition of congruent segments10.  CPCTC
  2. ABCDGiven: ABCD is a parallelogramProve: ∠A≅∠CStatementsReasons1.  ABCD is a parallelogramGiven2.  AB‾∥DC‾Definition of a parallelogram3.  ∠ABD≅∠CDB  4.  AD‾∥BC‾  5.  ∠ADB≅∠CBDAlternate interior angles of parallel lines are congruent6.  BD‾≅BD‾Reflexive property7.  △ABD≅△CDB  8.  ∠A≅∠CCPCTC3.  Alternate interior angles of parallel lines are congruent4.  Definition of a parallelogram7.  ASA
  3. DFGHEJGiven: DE‾≅GJ‾  DF‾≅GH‾  FE‾≅HJ‾Prove: GJ‾∥DE‾StatementsReasons1.  DE‾≅GJ‾Given2.  DF‾≅GH‾  3.  FE‾≅HJ‾Given4.  △DFE≅△GHJ  5.  ∠FDE≅∠HGJ  6.  GJ‾∥DE‾If corresponding angles are congruent, the lines are parallel2.  Given4.  SSS5.  CPCTC
  4. JKLMGiven: JK‾≅LK‾  KM‾⊥JL‾Prove: JM‾≅LM‾StatementsReasons1.  JK‾≅LK‾Given2.  KM‾⊥JL‾Given3.  ∠KMJ and ∠KML are right angles  4.  KM‾≅KM‾Reflexive property5.  △JKM≅△LKM  6.  JM‾≅LM‾  3.  Definition of perpendicular5.  HL6.  CPCTC

  1. Fill in every reason

    RSWTUGiven: W is the midpoint of RU‾ and ST‾Prove: RS‾∥TU‾Use the word bank.StatementsReasons1.  W is the midpoint of RU‾ and ST‾  2.  RW‾≅UW‾  3.  SW‾≅TW‾  4.  ∠RWS≅∠UWT  5.  △RSW≅△UTW  6.  ∠SRW≅∠TUW  7.  RS‾∥TU‾  Word bank: Given · Same-side interior angles of parallel lines are supplementary · Definition of a parallelogram · Corresponding angles of parallel lines are congruent · SAS · Definition of midpoint · Opposite angles of a parallelogram are congruent · CPCTC · If alternate interior angles are congruent, the lines are parallel · Vertical angles are congruent1.  Given2.  Definition of midpoint3.  Definition of midpoint4.  Vertical angles are congruent5.  SAS6.  CPCTC7.  If alternate interior angles are congruent, the lines are parallel
  2. QPRSGiven: SP‾⊥QP‾  SR‾⊥QR‾  SP‾≅SR‾Prove: QS‾ bisects ∠PQRUse the word bank.StatementsReasons1.  SP‾⊥QP‾  2.  SR‾⊥QR‾  3.  ∠QPS and ∠QRS are right angles  4.  SP‾≅SR‾  5.  QS‾≅QS‾  6.  △QPS≅△QRS  7.  ∠PQS≅∠RQS  8.  QS‾ bisects ∠PQR  Word bank: Reflexive property · Opposite angles of a parallelogram are congruent · Definition of an angle bisector · Given · The acute angles of a right triangle are complementary · AAS · Linear pairs are supplementary · Definition of perpendicular · HL · Angle addition postulate · CPCTC1.  Given2.  Given3.  Definition of perpendicular4.  Given5.  Reflexive property6.  HL7.  CPCTC8.  Definition of an angle bisector
  3. FGHIJKGiven: IJ‾∥FG‾  J is the midpoint of HG‾  IJ‾≅KG‾Prove: KJ‾∥FH‾Use the word bank.StatementsReasons1.  J is the midpoint of HG‾  2.  HJ‾≅JG‾  3.  IJ‾∥FG‾  4.  ∠HJI≅∠JGK  5.  IJ‾≅KG‾  6.  △HJI≅△JGK  7.  ∠IHJ≅∠KJG  8.  KJ‾∥FH‾  Word bank: Definition of midpoint · Corresponding angles of parallel lines are congruent · If both pairs of opposite sides are congruent, it is a parallelogram · Segment addition postulate · Definition of perpendicular · CPCTC · Given · SAS · The acute angles of a right triangle are complementary · If corresponding angles are congruent, the lines are parallel · Base angles of an isosceles triangle are congruent1.  Given2.  Definition of midpoint3.  Given4.  Corresponding angles of parallel lines are congruent5.  Given6.  SAS7.  CPCTC8.  If corresponding angles are congruent, the lines are parallel
  4. TYXZWGiven: ∠W≅∠Z  WY‾≅ZT‾  ZY‾∥WX‾Prove: TY‾≅YX‾Use the word bank.StatementsReasons1.  ∠W≅∠Z  2.  ZY‾∥WX‾  3.  ∠ZYT≅∠WXY  4.  WY‾≅ZT‾  5.  △ZTY≅△WYX  6.  TY‾≅YX‾  Word bank: Corresponding angles of parallel lines are congruent · AAS · Definition of a parallelogram · If alternate interior angles are congruent, the lines are parallel · Complements of the same angle are congruent · ASA · CPCTC · Given · Definition of a rectangle1.  Given2.  Given3.  Corresponding angles of parallel lines are congruent4.  Given5.  AAS6.  CPCTC

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How to do these

Given the triangles are congruent, why is ∠B≅∠C?

  1. The triangles were proved congruent on the line above.
  2. Angle B and angle C are matching parts of those triangles.

The answer is CPCTC.

Where students go wrong

Using CPCTC to prove the triangles congruent. It works the other way round: congruent triangles first, corresponding parts second.

Also called corresponding parts of congruent triangles, CPCTC worksheet, proofs using CPCTC or two column proofs CPCTC.

What you can put on this worksheet

Fill in the missing reasons
MNPQ
Given: ∠NQM≅∠NQP
  MQ‾≅PQ‾
Prove: MN‾≅PN‾
StatementsReasons1.  ∠NQM≅∠NQPGiven2.  MQ‾≅PQ‾Given3.  NQ‾≅NQ‾Reflexive property4.  △MNQ≅△PNQ  5.  MN‾≅PN‾  
→ Answer
4.  SAS
5.  CPCTC
Fill in every reason
MNPQ
Given: ∠NQM≅∠NQP
  MQ‾≅PQ‾
Prove: MN‾≅PN‾
Use the word bank.
StatementsReasons1.  ∠NQM≅∠NQP  2.  MQ‾≅PQ‾  3.  NQ‾≅NQ‾  4.  △MNQ≅△PNQ  5.  MN‾≅PN‾  
Word bank: Given · Opposite angles of a parallelogram are congruent · Definition of congruent segments · Reflexive property · Subtraction property · CPCTC · SAS · HL
→ Answer
1.  Given
2.  Given
3.  Reflexive property
4.  SAS
5.  CPCTC
Fill in the missing statements
MNPQ
Given: ∠NQM≅∠NQP
  MQ‾≅PQ‾
Prove: MN‾≅PN‾
StatementsReasons1.  ∠NQM≅∠NQPGiven2.  MQ‾≅PQ‾Given3.    Reflexive property4.    SAS5.  MN‾≅PN‾CPCTC
→ Answer
3.  NQ‾≅NQ‾
4.  △MNQ≅△PNQ
Write the whole proof
MNPQ
Given: ∠NQM≅∠NQP
  MQ‾≅PQ‾
Prove: MN‾≅PN‾
→ Answer
One possible proof:
StatementsReasons1.  ∠NQM≅∠NQPGiven2.  MQ‾≅PQ‾Given3.  NQ‾≅NQ‾Reflexive property4.  △MNQ≅△PNQSAS5.  MN‾≅PN‾CPCTC
Name the postulate that finishes the proof
KLMN
Given: KL‾≅KN‾
  ML‾≅MN‾
Prove: △KLM≅△KNM
Which congruence postulate or theorem proves the triangles congruent?
StatementsReasons1.  KL‾≅KN‾Given2.  ML‾≅MN‾Given3.  KM‾≅KM‾Reflexive property4.  △KLM≅△KNM  
→ Answer
SSS

Questions about these worksheets

Yes. Every cpctc 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 cpctc 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.

Using CPCTC to prove the triangles congruent. It works the other way round: congruent triangles first, corresponding parts second.