Worksheets · Geometry

Proving triangles congruent worksheet

Every one of these proofs ends the same way: line up three matching parts in the right pattern and name the postulate. The work is getting those three parts. A shared side is congruent to itself by the reflexive property, a midpoint splits a segment into two congruent halves, vertical angles are congruent, and parallel lines hand you congruent alternate interior angles. Collect three, check the pattern, and write SSS, SAS, ASA, AAS or HL.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Proving Triangles Congruent

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

    PMLNKGiven: ∠K≅∠N  KM‾≅NP‾  KL‾∥NM‾Prove: △NPM≅△KMLStatementsReasons1.  ∠K≅∠NGiven2.  KM‾≅NP‾Given3.  KL‾∥NM‾  4.  ∠NMP≅∠KLM  5.  △NPM≅△KML  
  2. TVSUGiven: SV‾⊥TV‾  TU‾⊥TV‾  ST‾≅UV‾Prove: △STV≅△UVTStatementsReasons1.  SV‾⊥TV‾Given2.  TU‾⊥TV‾  3.  ∠SVT and ∠UTV are right angles  4.  ST‾≅UV‾Given5.  TV‾≅VT‾  6.  △STV≅△UVTHL
  3. PQRSTGiven: PS‾≅QR‾  △STR is isosceles with base SR‾  △PQT is isosceles with base PQ‾Prove: △PTS≅△QTRStatementsReasons1.  PS‾≅QR‾Given2.  △STR is isosceles with base SR‾Given3.  ST‾≅RT‾  4.  △PQT is isosceles with base PQ‾  5.  PT‾≅QT‾  6.  △PTS≅△QTRSSS
  4. JFHEDGGiven: DE‾∥HG‾  ∠F≅∠J  DE‾≅HG‾Prove: △DEJ≅△GHFStatementsReasons1.  DE‾∥HG‾Given2.  ∠DEJ≅∠GHF  3.  ∠F≅∠J  4.  DE‾≅HG‾  5.  △DEJ≅△GHFAAS

  1. Fill in every reason

    RSWUTGiven: RT‾≅UT‾  SR‾≅WU‾Prove: △RST≅△UWTUse the word bank.StatementsReasons1.  RT‾≅UT‾  2.  ∠R≅∠U  3.  SR‾≅WU‾  4.  △RST≅△UWT  Word bank: Definition of a median · Halves of congruent angles are congruent · Substitution · All right angles are congruent · SAS · Given · Base angles of an isosceles triangle are congruent
  2. MNPQRGiven: MN‾∥PQ‾  R is the midpoint of MQ‾Prove: △MNR≅△QPRUse the word bank.StatementsReasons1.  MN‾∥PQ‾  2.  ∠NMR≅∠PQR  3.  R is the midpoint of MQ‾  4.  MR‾≅QR‾  5.  ∠MRN≅∠QRP  6.  △MNR≅△QPR  Word bank: Given · Vertical angles are congruent · Subtraction property · Opposite sides of a parallelogram are congruent · Definition of midpoint · ASA · Definition of an isosceles triangle · Alternate interior angles of parallel lines are congruent · Transitive property
  3. JGHKLMGiven: L is the midpoint of JH‾  GJ‾∥ML‾  KL‾∥GH‾Prove: △KJL≅△MLHUse the word bank.StatementsReasons1.  L is the midpoint of JH‾  2.  JL‾≅LH‾  3.  GJ‾∥ML‾  4.  ∠KJL≅∠MLH  5.  KL‾∥GH‾  6.  ∠JLK≅∠LHM  7.  △KJL≅△MLH  Word bank: All right angles are congruent · Definition of a rectangle · ASA · Opposite sides of a rectangle are congruent · Definition of midpoint · Angle addition postulate · Reflexive property · Definition of congruent segments · Corresponding angles of parallel lines are congruent · Given
  4. TUWYGiven: YW‾ is the perpendicular bisector of TU‾Prove: △TWY≅△UWYUse the word bank.StatementsReasons1.  YW‾ is the perpendicular bisector of TU‾  2.  TW‾≅UW‾  3.  ∠YWT and ∠YWU are right angles  4.  ∠YWT≅∠YWU  5.  YW‾≅YW‾  6.  △TWY≅△UWY  Word bank: Definition of a perpendicular bisector · Given · SAS · CPCTC · Opposite sides of a rectangle are congruent · Vertical angles are congruent · Angle addition postulate · Reflexive property · All right angles are congruent

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

Sigma Prep · sigmaprep.io/worksheets

Name

Proving Triangles CongruentAnswer 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

    PMLNKGiven: ∠K≅∠N  KM‾≅NP‾  KL‾∥NM‾Prove: △NPM≅△KMLStatementsReasons1.  ∠K≅∠NGiven2.  KM‾≅NP‾Given3.  KL‾∥NM‾  4.  ∠NMP≅∠KLM  5.  △NPM≅△KML  3.  Given4.  Corresponding angles of parallel lines are congruent5.  AAS
  2. TVSUGiven: SV‾⊥TV‾  TU‾⊥TV‾  ST‾≅UV‾Prove: △STV≅△UVTStatementsReasons1.  SV‾⊥TV‾Given2.  TU‾⊥TV‾  3.  ∠SVT and ∠UTV are right angles  4.  ST‾≅UV‾Given5.  TV‾≅VT‾  6.  △STV≅△UVTHL2.  Given3.  Definition of perpendicular5.  Reflexive property
  3. PQRSTGiven: PS‾≅QR‾  △STR is isosceles with base SR‾  △PQT is isosceles with base PQ‾Prove: △PTS≅△QTRStatementsReasons1.  PS‾≅QR‾Given2.  △STR is isosceles with base SR‾Given3.  ST‾≅RT‾  4.  △PQT is isosceles with base PQ‾  5.  PT‾≅QT‾  6.  △PTS≅△QTRSSS3.  Definition of an isosceles triangle4.  Given5.  Definition of an isosceles triangle
  4. JFHEDGGiven: DE‾∥HG‾  ∠F≅∠J  DE‾≅HG‾Prove: △DEJ≅△GHFStatementsReasons1.  DE‾∥HG‾Given2.  ∠DEJ≅∠GHF  3.  ∠F≅∠J  4.  DE‾≅HG‾  5.  △DEJ≅△GHFAAS2.  Alternate interior angles of parallel lines are congruent3.  Given4.  Given

  1. Fill in every reason

    RSWUTGiven: RT‾≅UT‾  SR‾≅WU‾Prove: △RST≅△UWTUse the word bank.StatementsReasons1.  RT‾≅UT‾  2.  ∠R≅∠U  3.  SR‾≅WU‾  4.  △RST≅△UWT  Word bank: Definition of a median · Halves of congruent angles are congruent · Substitution · All right angles are congruent · SAS · Given · Base angles of an isosceles triangle are congruent1.  Given2.  Base angles of an isosceles triangle are congruent3.  Given4.  SAS
  2. MNPQRGiven: MN‾∥PQ‾  R is the midpoint of MQ‾Prove: △MNR≅△QPRUse the word bank.StatementsReasons1.  MN‾∥PQ‾  2.  ∠NMR≅∠PQR  3.  R is the midpoint of MQ‾  4.  MR‾≅QR‾  5.  ∠MRN≅∠QRP  6.  △MNR≅△QPR  Word bank: Given · Vertical angles are congruent · Subtraction property · Opposite sides of a parallelogram are congruent · Definition of midpoint · ASA · Definition of an isosceles triangle · Alternate interior angles of parallel lines are congruent · Transitive property1.  Given2.  Alternate interior angles of parallel lines are congruent3.  Given4.  Definition of midpoint5.  Vertical angles are congruent6.  ASA
  3. JGHKLMGiven: L is the midpoint of JH‾  GJ‾∥ML‾  KL‾∥GH‾Prove: △KJL≅△MLHUse the word bank.StatementsReasons1.  L is the midpoint of JH‾  2.  JL‾≅LH‾  3.  GJ‾∥ML‾  4.  ∠KJL≅∠MLH  5.  KL‾∥GH‾  6.  ∠JLK≅∠LHM  7.  △KJL≅△MLH  Word bank: All right angles are congruent · Definition of a rectangle · ASA · Opposite sides of a rectangle are congruent · Definition of midpoint · Angle addition postulate · Reflexive property · Definition of congruent segments · Corresponding angles of parallel lines are congruent · Given1.  Given2.  Definition of midpoint3.  Given4.  Corresponding angles of parallel lines are congruent5.  Given6.  Corresponding angles of parallel lines are congruent7.  ASA
  4. TUWYGiven: YW‾ is the perpendicular bisector of TU‾Prove: △TWY≅△UWYUse the word bank.StatementsReasons1.  YW‾ is the perpendicular bisector of TU‾  2.  TW‾≅UW‾  3.  ∠YWT and ∠YWU are right angles  4.  ∠YWT≅∠YWU  5.  YW‾≅YW‾  6.  △TWY≅△UWY  Word bank: Definition of a perpendicular bisector · Given · SAS · CPCTC · Opposite sides of a rectangle are congruent · Vertical angles are congruent · Angle addition postulate · Reflexive property · All right angles are congruent1.  Given2.  Definition of a perpendicular bisector3.  Definition of a perpendicular bisector4.  All right angles are congruent5.  Reflexive property6.  SAS

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

How to do these

Given AB‾≅CD‾ and BC‾≅DA‾. Give the reason for the last line: △ABC≅△CDA

  1. Two pairs of sides are given congruent.
  2. The third pair is the shared side AC, congruent to itself by the reflexive property.
  3. Three pairs of sides.

The answer is SSS.

Where students go wrong

Marking a pattern that does not prove anything. Two sides and an angle that is not between them is SSA, and SSA proves nothing at all.

Also called triangle congruence proofs, two column proofs triangles, SSS SAS ASA AAS proofs or congruent triangle proofs.

What you can put on this worksheet

Fill in the missing reasons
MNPQRS
Given: ∠M≅∠Q
  ∠N≅∠R
  NP‾≅RS‾
Prove: △MNP≅△QRS
StatementsReasons1.  ∠M≅∠QGiven2.  ∠N≅∠RGiven3.  NP‾≅RS‾  4.  △MNP≅△QRS  
→ Answer
3.  Given
4.  AAS
Fill in every reason
MNPQRS
Given: ∠M≅∠Q
  ∠N≅∠R
  NP‾≅RS‾
Prove: △MNP≅△QRS
Use the word bank.
StatementsReasons1.  ∠M≅∠Q  2.  ∠N≅∠R  3.  NP‾≅RS‾  4.  △MNP≅△QRS  
Word bank: Given · Definition of congruent segments · HL · AAS · Addition property · SSS · Complements of the same angle are congruent
→ Answer
1.  Given
2.  Given
3.  Given
4.  AAS
Fill in the missing statements
MNPQRS
Given: ∠M≅∠Q
  ∠N≅∠R
  NP‾≅RS‾
Prove: △MNP≅△QRS
StatementsReasons1.  ∠M≅∠QGiven2.    Given3.    Given4.  △MNP≅△QRSAAS
→ Answer
2.  ∠N≅∠R
3.  NP‾≅RS‾
Write the whole proof
MNPQRS
Given: ∠M≅∠Q
  ∠N≅∠R
  NP‾≅RS‾
Prove: △MNP≅△QRS
→ Answer
One possible proof:
StatementsReasons1.  ∠M≅∠QGiven2.  ∠N≅∠RGiven3.  NP‾≅RS‾Given4.  △MNP≅△QRSAAS
Name the postulate that finishes the proof
KMLNQP
Given: KL‾≅NP‾
  ∠LKM≅∠PNQ
  KM‾≅NQ‾
Prove: △KLM≅△NPQ
Which congruence postulate or theorem proves the triangles congruent?
StatementsReasons1.  KL‾≅NP‾Given2.  ∠LKM≅∠PNQGiven3.  KM‾≅NQ‾Given4.  △KLM≅△NPQ  
→ Answer
SAS

Questions about these worksheets

Yes. Every proving triangles congruent 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 triangles congruent 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.

Marking a pattern that does not prove anything. Two sides and an angle that is not between them is SSA, and SSA proves nothing at all.