Worksheets · Geometry

Segment and angle proofs worksheet

These are the first proofs a geometry course asks for, and they run on properties rather than theorems. The segment and angle addition postulates let you split a whole into parts. A midpoint gives two congruent halves, a bisector does the same for an angle, and anything is congruent to itself by the reflexive property. Adding or subtracting congruent pieces from congruent wholes keeps them congruent. Some proofs use vertical angles, or the angle pairs a transversal makes with parallel lines, instead.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Segment and Angle 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

    12Given: ∠1≅∠2Prove: ∠1 and ∠2 are right anglesStatementsReasons1.  ∠1≅∠2Given2.  ∠1 and ∠2 form a linear pairDefinition of a linear pair3.  ∠1 and ∠2 are supplementary  4.  m∠1+m∠2=180∘Definition of supplementary angles5.  m∠1=m∠2  6.  2m∠1=180∘Substitution7.  m∠1=90∘Division property8.  m∠2=90∘Substitution9.  ∠1 and ∠2 are right angles  
  2. 1234TUSYWXVZGiven: m∠1=(x+12)∘  m∠3=(2x+3)∘  m∠STU=75∘Prove: x=20StatementsReasons1.  m∠1=(x+12)∘Given2.  m∠3=(2x+3)∘Given3.  m∠STU=75∘  4.  m∠1+m∠3=m∠STUAngle addition postulate5.  x+12+2x+3=75Substitution6.  3x+15=75  7.  3x=60  8.  x=20Division property
  3. KMLNGiven: NL→ bisects ∠KNM  ∠K and ∠KNL are complementary  ∠M and ∠MNL are complementaryProve: ∠K≅∠MStatementsReasons1.  NL→ bisects ∠KNMGiven2.  ∠KNL≅∠MNL  3.  ∠K and ∠KNL are complementary  4.  ∠M and ∠MNL are complementaryGiven5.  ∠K≅∠M  
  4. 123Given: ∠1 and ∠2 are complementaryProve: ∠2 and ∠3 are complementaryStatementsReasons1.  ∠1 and ∠2 are complementaryGiven2.  m∠1+m∠2=90∘Definition of complementary angles3.  ∠1≅∠3  4.  m∠1=m∠3  5.  m∠3+m∠2=90∘  6.  ∠2 and ∠3 are complementaryDefinition of complementary angles

  1. Fill in every reason

    12345678ℓmtGiven: ℓ∥mProve: ∠3 and ∠5 are supplementaryUse the word bank.StatementsReasons1.  ℓ∥m  2.  ∠3≅∠6  3.  m∠3=m∠6  4.  ∠6 and ∠5 are supplementary  5.  m∠6+m∠5=180∘  6.  m∠3+m∠5=180∘  7.  ∠3 and ∠5 are supplementary  Word bank: Definition of midpoint · Given · Addition property · Definition of an angle bisector · Linear pairs are supplementary · Alternate interior angles of parallel lines are congruent · Definition of congruent angles · Definition of a linear pair · Substitution · Definition of supplementary angles
  2. MNPGiven: N is the midpoint of MP‾  MN=3x−5  NP=x+1Prove: MP=8Use the word bank.StatementsReasons1.  N is the midpoint of MP‾  2.  MN=3x−5  3.  NP=x+1  4.  MN=NP  5.  3x−5=x+1  6.  2x−5=1  7.  2x=6  8.  x=3  9.  MN=4  10.  NP=4  11.  MN+NP=MP  12.  4+4=MP  13.  MP=8  Word bank: Alternate interior angles of parallel lines are congruent · Definition of complementary angles · Simplify · Definition of congruent segments · Division property · Supplements of congruent angles are congruent · Given · Distributive property · Definition of an angle bisector · Segment addition postulate · Definition of midpoint · Subtraction property · Definition of supplementary angles · Adjacent angles that form a right angle are complementary · Substitution · Addition property
  3. DEFGGiven: ∠D and ∠E are supplementary  ∠F and ∠G are supplementary  ∠E≅∠FProve: ∠D≅∠GUse the word bank.StatementsReasons1.  ∠D and ∠E are supplementary  2.  ∠F and ∠G are supplementary  3.  ∠E≅∠F  4.  m∠D+m∠E=180∘  5.  m∠F+m∠G=180∘  6.  m∠D+m∠E=m∠F+m∠G  7.  m∠E=m∠F  8.  m∠D=m∠G  9.  ∠D≅∠G  Word bank: Supplements of the same angle are congruent · Definition of supplementary angles · Segment addition postulate · Distributive property · Alternate exterior angles of parallel lines are congruent · Division property · Given · Definition of a segment bisector · Subtraction property · Substitution · Angle addition postulate · Definition of congruent angles
  4. JKLMNPGiven: JK‾≅NP‾  LM‾≅NP‾Prove: JL‾≅KM‾Use the word bank.StatementsReasons1.  JK‾≅NP‾  2.  LM‾≅NP‾  3.  JK‾≅LM‾  4.  JK=LM  5.  JK+KL=LM+KL  6.  JK+KL=JL and KL+LM=KM  7.  JL=KM  8.  JL‾≅KM‾  Word bank: Substitution · Transitive property · Given · Corresponding angles of parallel lines are congruent · Supplements of the same angle are congruent · Simplify · Definition of a linear pair · Definition of congruent segments · Segment addition postulate · Angle addition postulate · Addition property

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

Sigma Prep · sigmaprep.io/worksheets

Name

Segment and Angle 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

    12Given: ∠1≅∠2Prove: ∠1 and ∠2 are right anglesStatementsReasons1.  ∠1≅∠2Given2.  ∠1 and ∠2 form a linear pairDefinition of a linear pair3.  ∠1 and ∠2 are supplementary  4.  m∠1+m∠2=180∘Definition of supplementary angles5.  m∠1=m∠2  6.  2m∠1=180∘Substitution7.  m∠1=90∘Division property8.  m∠2=90∘Substitution9.  ∠1 and ∠2 are right angles  3.  Linear pairs are supplementary5.  Definition of congruent angles9.  Definition of a right angle
  2. 1234TUSYWXVZGiven: m∠1=(x+12)∘  m∠3=(2x+3)∘  m∠STU=75∘Prove: x=20StatementsReasons1.  m∠1=(x+12)∘Given2.  m∠3=(2x+3)∘Given3.  m∠STU=75∘  4.  m∠1+m∠3=m∠STUAngle addition postulate5.  x+12+2x+3=75Substitution6.  3x+15=75  7.  3x=60  8.  x=20Division property3.  Given6.  Simplify7.  Subtraction property
  3. KMLNGiven: NL→ bisects ∠KNM  ∠K and ∠KNL are complementary  ∠M and ∠MNL are complementaryProve: ∠K≅∠MStatementsReasons1.  NL→ bisects ∠KNMGiven2.  ∠KNL≅∠MNL  3.  ∠K and ∠KNL are complementary  4.  ∠M and ∠MNL are complementaryGiven5.  ∠K≅∠M  2.  Definition of an angle bisector3.  Given5.  Complements of congruent angles are congruent
  4. 123Given: ∠1 and ∠2 are complementaryProve: ∠2 and ∠3 are complementaryStatementsReasons1.  ∠1 and ∠2 are complementaryGiven2.  m∠1+m∠2=90∘Definition of complementary angles3.  ∠1≅∠3  4.  m∠1=m∠3  5.  m∠3+m∠2=90∘  6.  ∠2 and ∠3 are complementaryDefinition of complementary angles3.  Vertical angles are congruent4.  Definition of congruent angles5.  Substitution

  1. Fill in every reason

    12345678ℓmtGiven: ℓ∥mProve: ∠3 and ∠5 are supplementaryUse the word bank.StatementsReasons1.  ℓ∥m  2.  ∠3≅∠6  3.  m∠3=m∠6  4.  ∠6 and ∠5 are supplementary  5.  m∠6+m∠5=180∘  6.  m∠3+m∠5=180∘  7.  ∠3 and ∠5 are supplementary  Word bank: Definition of midpoint · Given · Addition property · Definition of an angle bisector · Linear pairs are supplementary · Alternate interior angles of parallel lines are congruent · Definition of congruent angles · Definition of a linear pair · Substitution · Definition of supplementary angles1.  Given2.  Alternate interior angles of parallel lines are congruent3.  Definition of congruent angles4.  Linear pairs are supplementary5.  Definition of supplementary angles6.  Substitution7.  Definition of supplementary angles
  2. MNPGiven: N is the midpoint of MP‾  MN=3x−5  NP=x+1Prove: MP=8Use the word bank.StatementsReasons1.  N is the midpoint of MP‾  2.  MN=3x−5  3.  NP=x+1  4.  MN=NP  5.  3x−5=x+1  6.  2x−5=1  7.  2x=6  8.  x=3  9.  MN=4  10.  NP=4  11.  MN+NP=MP  12.  4+4=MP  13.  MP=8  Word bank: Alternate interior angles of parallel lines are congruent · Definition of complementary angles · Simplify · Definition of congruent segments · Division property · Supplements of congruent angles are congruent · Given · Distributive property · Definition of an angle bisector · Segment addition postulate · Definition of midpoint · Subtraction property · Definition of supplementary angles · Adjacent angles that form a right angle are complementary · Substitution · Addition property1.  Given2.  Given3.  Given4.  Definition of midpoint5.  Substitution6.  Subtraction property7.  Addition property8.  Division property9.  Substitution10.  Substitution11.  Segment addition postulate12.  Substitution13.  Simplify
  3. DEFGGiven: ∠D and ∠E are supplementary  ∠F and ∠G are supplementary  ∠E≅∠FProve: ∠D≅∠GUse the word bank.StatementsReasons1.  ∠D and ∠E are supplementary  2.  ∠F and ∠G are supplementary  3.  ∠E≅∠F  4.  m∠D+m∠E=180∘  5.  m∠F+m∠G=180∘  6.  m∠D+m∠E=m∠F+m∠G  7.  m∠E=m∠F  8.  m∠D=m∠G  9.  ∠D≅∠G  Word bank: Supplements of the same angle are congruent · Definition of supplementary angles · Segment addition postulate · Distributive property · Alternate exterior angles of parallel lines are congruent · Division property · Given · Definition of a segment bisector · Subtraction property · Substitution · Angle addition postulate · Definition of congruent angles1.  Given2.  Given3.  Given4.  Definition of supplementary angles5.  Definition of supplementary angles6.  Substitution7.  Definition of congruent angles8.  Subtraction property9.  Definition of congruent angles
  4. JKLMNPGiven: JK‾≅NP‾  LM‾≅NP‾Prove: JL‾≅KM‾Use the word bank.StatementsReasons1.  JK‾≅NP‾  2.  LM‾≅NP‾  3.  JK‾≅LM‾  4.  JK=LM  5.  JK+KL=LM+KL  6.  JK+KL=JL and KL+LM=KM  7.  JL=KM  8.  JL‾≅KM‾  Word bank: Substitution · Transitive property · Given · Corresponding angles of parallel lines are congruent · Supplements of the same angle are congruent · Simplify · Definition of a linear pair · Definition of congruent segments · Segment addition postulate · Angle addition postulate · Addition property1.  Given2.  Given3.  Transitive property4.  Definition of congruent segments5.  Addition property6.  Segment addition postulate7.  Substitution8.  Definition of congruent segments

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

How to do these

Given AB‾≅CD‾, prove AC‾≅BD‾

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

The answer is Addition property.

Where students go wrong

Skipping the addition postulate line. Going straight from the Given to the conclusion leaves out the step that actually justifies it.

Also called segment addition proofs, angle addition postulate proofs, beginning proofs or intro to proofs worksheet.

What you can put on this worksheet

Fill in the missing reasons
KLMN
Given: L is the midpoint of KM‾
  KL‾≅MN‾
Prove: LM‾≅MN‾
StatementsReasons1.  L is the midpoint of KM‾Given2.  KL‾≅LM‾Definition of midpoint3.  KL‾≅MN‾  4.  LM‾≅MN‾  
→ Answer
3.  Given
4.  Transitive property
Fill in every reason
KLMN
Given: L is the midpoint of KM‾
  KL‾≅MN‾
Prove: LM‾≅MN‾
Use the word bank.
StatementsReasons1.  L is the midpoint of KM‾  2.  KL‾≅LM‾  3.  KL‾≅MN‾  4.  LM‾≅MN‾  
Word bank: Definition of midpoint · Multiplication property · Definition of supplementary angles · Distributive property · Transitive property · Linear pairs are supplementary · Given
→ Answer
1.  Given
2.  Definition of midpoint
3.  Given
4.  Transitive property
Fill in the missing statements
KLMN
Given: L is the midpoint of KM‾
  KL‾≅MN‾
Prove: LM‾≅MN‾
StatementsReasons1.  L is the midpoint of KM‾Given2.    Definition of midpoint3.    Given4.  LM‾≅MN‾Transitive property
→ Answer
2.  KL‾≅LM‾
3.  KL‾≅MN‾
Write the whole proof
KLMN
Given: L is the midpoint of KM‾
  KL‾≅MN‾
Prove: LM‾≅MN‾
→ Answer
One possible proof:
StatementsReasons1.  L is the midpoint of KM‾Given2.  KL‾≅LM‾Definition of midpoint3.  KL‾≅MN‾Given4.  LM‾≅MN‾Transitive property

Questions about these worksheets

Yes. Every segment and angle 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 segment and angle 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 4: fill in the missing reasons, fill in every reason, fill in the missing statements and write the whole proof. Tick as many as you want and set how many of each, or let it spread them evenly.

Skipping the addition postulate line. Going straight from the Given to the conclusion leaves out the step that actually justifies it.