Worksheets · Geometry

Circle proofs worksheet

Circle proofs start from one fact that is always available: every radius of a circle is congruent to every other. That single line turns most circle diagrams into isosceles triangles, which brings the base angle theorem with it. Two more theorems do the rest of the work. A radius drawn perpendicular to a chord bisects it, and inscribed angles that intercept the same arc are congruent.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Circle 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

    OMNKLPGiven: circle O with diameter MN‾ and chord KL‾  MN‾⊥KL‾ at PProve: P is the midpoint of KL‾StatementsReasons1.  circle O with diameter MN‾ and chord KL‾Given2.  MN‾⊥KL‾ at P  3.  ∠OPK and ∠OPL are right anglesDefinition of perpendicular4.  OK‾≅OL‾All radii of a circle are congruent5.  OP‾≅OP‾  6.  △OPK≅△OPLHL7.  KP‾≅LP‾CPCTC8.  P is the midpoint of KL‾  
  2. MNPQGiven: circles M and N meet at P and QProve: △MPN≅△MQNStatementsReasons1.  circles M and N meet at P and QGiven2.  MP‾≅MQ‾  3.  NP‾≅NQ‾  4.  MN‾≅MN‾  5.  △MPN≅△MQNSSS
  3. OWXYZTGiven: chords WX‾ and YZ‾ of circle O meet at TProve: WT⋅XT=YT⋅ZTStatementsReasons1.  chords WX‾ and YZ‾ of circle O meet at TGiven2.  ∠YWX≅∠YZXInscribed angles that intercept the same arc are congruent3.  ∠WYZ≅∠WXZ  4.  △WYT∼△ZXT  5.  WTZT=YTXTCorresponding sides of similar triangles are proportional6.  WT⋅XT=YT⋅ZT  
  4. ORSTUGiven: RSTU is inscribed in circle OProve: ∠RST and ∠RUT are supplementaryStatementsReasons1.  RSTU is inscribed in circle OGiven2.  m∠RST=12mRUT⌢Inscribed angle theorem3.  m∠RUT=12mRST⌢Inscribed angle theorem4.  mRUT⌢+mRST⌢=360∘Arc addition postulate5.  12mRUT⌢+12mRST⌢=180∘  6.  m∠RST+m∠RUT=180∘  7.  ∠RST and ∠RUT are supplementary  

  1. Fill in every reason

    OSTUVGiven: S,T,U, and V lie on circle O  ST‾∥VU‾Prove: SV⌢≅TU⌢Use the word bank.StatementsReasons1.  S,T,U, and V lie on circle O  2.  ST‾∥VU‾  3.  ∠STV≅∠UVT  4.  m∠STV=m∠UVT  5.  mSV⌢=2m∠STV  6.  mTU⌢=2m∠UVT  7.  mSV⌢=mTU⌢  8.  SV⌢≅TU⌢  Word bank: Distributive property · Halves of congruent segments are congruent · SAS · Inscribed angle theorem · Vertical angles are congruent · Definition of congruent arcs · Substitution · Given · Definition of congruent angles · Alternate interior angles of parallel lines are congruent · All radii of a circle are congruent
  2. OPQRSTGiven: PQ‾ meets circle O at S and Q  PR‾ meets circle O at T and RProve: m∠QPR=12(mQR⌢−mST⌢)Use the word bank.StatementsReasons1.  PQ‾ meets circle O at S and Q  2.  PR‾ meets circle O at T and R  3.  m∠QTR=12mQR⌢  4.  m∠PQT=12mST⌢  5.  m∠QPR+m∠PQT=m∠QTR  6.  m∠QPR=m∠QTR−m∠PQT  7.  m∠QPR=12(mQR⌢−mST⌢)  Word bank: Exterior angle theorem · Substitution · All right angles are congruent · Definition of midpoint · Inscribed angle theorem · Inscribed angles that intercept the same arc are congruent · Subtraction property · Given · All radii of a circle are congruent · Definition of an isosceles triangle
  3. OFGHIGiven: IG‾ is tangent to circle O at G  F and H lie on circle O  H is the midpoint of FHG⌢Prove: ∠FGH≅∠HGIUse the word bank.StatementsReasons1.  IG‾ is tangent to circle O at G  2.  F and H lie on circle O  3.  H is the midpoint of FHG⌢  4.  FH⌢≅HG⌢  5.  mFH⌢=mHG⌢  6.  m∠FGH=12mFH⌢  7.  m∠HGI=12mHG⌢  8.  m∠FGH=m∠HGI  9.  ∠FGH≅∠HGI  Word bank: HL · Definition of congruent angles · All right angles are congruent · ASA · An angle inscribed in a semicircle is a right angle · Segment addition postulate · Definition of the midpoint of an arc · Substitution · Definition of congruent arcs · A tangent-chord angle is half its intercepted arc · Inscribed angle theorem · Given
  4. ORSTUGiven: RS‾ is tangent to circle O at S  secant RU‾ meets circle O at T and UProve: △RST∼△RUSUse the word bank.StatementsReasons1.  RS‾ is tangent to circle O at S  2.  secant RU‾ meets circle O at T and U  3.  ∠R≅∠R  4.  m∠RST=12mST⌢  5.  m∠RUS=12mST⌢  6.  m∠RST=m∠RUS  7.  ∠RST≅∠RUS  8.  △RST∼△RUS  Word bank: AA similarity · Definition of congruent angles · Definition of the midpoint of an arc · Substitution · A tangent-chord angle is half its intercepted arc · Inscribed angle theorem · Reflexive property · The acute angles of a right triangle are complementary · Two points determine a line · Given · Sides opposite congruent angles are congruent

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

Sigma Prep · sigmaprep.io/worksheets

Name

Circle 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

    OMNKLPGiven: circle O with diameter MN‾ and chord KL‾  MN‾⊥KL‾ at PProve: P is the midpoint of KL‾StatementsReasons1.  circle O with diameter MN‾ and chord KL‾Given2.  MN‾⊥KL‾ at P  3.  ∠OPK and ∠OPL are right anglesDefinition of perpendicular4.  OK‾≅OL‾All radii of a circle are congruent5.  OP‾≅OP‾  6.  △OPK≅△OPLHL7.  KP‾≅LP‾CPCTC8.  P is the midpoint of KL‾  2.  Given5.  Reflexive property8.  Definition of midpoint
  2. MNPQGiven: circles M and N meet at P and QProve: △MPN≅△MQNStatementsReasons1.  circles M and N meet at P and QGiven2.  MP‾≅MQ‾  3.  NP‾≅NQ‾  4.  MN‾≅MN‾  5.  △MPN≅△MQNSSS2.  All radii of a circle are congruent3.  All radii of a circle are congruent4.  Reflexive property
  3. OWXYZTGiven: chords WX‾ and YZ‾ of circle O meet at TProve: WT⋅XT=YT⋅ZTStatementsReasons1.  chords WX‾ and YZ‾ of circle O meet at TGiven2.  ∠YWX≅∠YZXInscribed angles that intercept the same arc are congruent3.  ∠WYZ≅∠WXZ  4.  △WYT∼△ZXT  5.  WTZT=YTXTCorresponding sides of similar triangles are proportional6.  WT⋅XT=YT⋅ZT  3.  Inscribed angles that intercept the same arc are congruent4.  AA similarity6.  Cross products property
  4. ORSTUGiven: RSTU is inscribed in circle OProve: ∠RST and ∠RUT are supplementaryStatementsReasons1.  RSTU is inscribed in circle OGiven2.  m∠RST=12mRUT⌢Inscribed angle theorem3.  m∠RUT=12mRST⌢Inscribed angle theorem4.  mRUT⌢+mRST⌢=360∘Arc addition postulate5.  12mRUT⌢+12mRST⌢=180∘  6.  m∠RST+m∠RUT=180∘  7.  ∠RST and ∠RUT are supplementary  5.  Multiplication property6.  Substitution7.  Definition of supplementary angles

  1. Fill in every reason

    OSTUVGiven: S,T,U, and V lie on circle O  ST‾∥VU‾Prove: SV⌢≅TU⌢Use the word bank.StatementsReasons1.  S,T,U, and V lie on circle O  2.  ST‾∥VU‾  3.  ∠STV≅∠UVT  4.  m∠STV=m∠UVT  5.  mSV⌢=2m∠STV  6.  mTU⌢=2m∠UVT  7.  mSV⌢=mTU⌢  8.  SV⌢≅TU⌢  Word bank: Distributive property · Halves of congruent segments are congruent · SAS · Inscribed angle theorem · Vertical angles are congruent · Definition of congruent arcs · Substitution · Given · Definition of congruent angles · Alternate interior angles of parallel lines are congruent · All radii of a circle are congruent1.  Given2.  Given3.  Alternate interior angles of parallel lines are congruent4.  Definition of congruent angles5.  Inscribed angle theorem6.  Inscribed angle theorem7.  Substitution8.  Definition of congruent arcs
  2. OPQRSTGiven: PQ‾ meets circle O at S and Q  PR‾ meets circle O at T and RProve: m∠QPR=12(mQR⌢−mST⌢)Use the word bank.StatementsReasons1.  PQ‾ meets circle O at S and Q  2.  PR‾ meets circle O at T and R  3.  m∠QTR=12mQR⌢  4.  m∠PQT=12mST⌢  5.  m∠QPR+m∠PQT=m∠QTR  6.  m∠QPR=m∠QTR−m∠PQT  7.  m∠QPR=12(mQR⌢−mST⌢)  Word bank: Exterior angle theorem · Substitution · All right angles are congruent · Definition of midpoint · Inscribed angle theorem · Inscribed angles that intercept the same arc are congruent · Subtraction property · Given · All radii of a circle are congruent · Definition of an isosceles triangle1.  Given2.  Given3.  Inscribed angle theorem4.  Inscribed angle theorem5.  Exterior angle theorem6.  Subtraction property7.  Substitution
  3. OFGHIGiven: IG‾ is tangent to circle O at G  F and H lie on circle O  H is the midpoint of FHG⌢Prove: ∠FGH≅∠HGIUse the word bank.StatementsReasons1.  IG‾ is tangent to circle O at G  2.  F and H lie on circle O  3.  H is the midpoint of FHG⌢  4.  FH⌢≅HG⌢  5.  mFH⌢=mHG⌢  6.  m∠FGH=12mFH⌢  7.  m∠HGI=12mHG⌢  8.  m∠FGH=m∠HGI  9.  ∠FGH≅∠HGI  Word bank: HL · Definition of congruent angles · All right angles are congruent · ASA · An angle inscribed in a semicircle is a right angle · Segment addition postulate · Definition of the midpoint of an arc · Substitution · Definition of congruent arcs · A tangent-chord angle is half its intercepted arc · Inscribed angle theorem · Given1.  Given2.  Given3.  Given4.  Definition of the midpoint of an arc5.  Definition of congruent arcs6.  Inscribed angle theorem7.  A tangent-chord angle is half its intercepted arc8.  Substitution9.  Definition of congruent angles
  4. ORSTUGiven: RS‾ is tangent to circle O at S  secant RU‾ meets circle O at T and UProve: △RST∼△RUSUse the word bank.StatementsReasons1.  RS‾ is tangent to circle O at S  2.  secant RU‾ meets circle O at T and U  3.  ∠R≅∠R  4.  m∠RST=12mST⌢  5.  m∠RUS=12mST⌢  6.  m∠RST=m∠RUS  7.  ∠RST≅∠RUS  8.  △RST∼△RUS  Word bank: AA similarity · Definition of congruent angles · Definition of the midpoint of an arc · Substitution · A tangent-chord angle is half its intercepted arc · Inscribed angle theorem · Reflexive property · The acute angles of a right triangle are complementary · Two points determine a line · Given · Sides opposite congruent angles are congruent1.  Given2.  Given3.  Reflexive property4.  A tangent-chord angle is half its intercepted arc5.  Inscribed angle theorem6.  Substitution7.  Definition of congruent angles8.  AA similarity

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

How to do these

Given circle O, why is △OAB isosceles?

  1. OA and OB are both radii of circle O.
  2. All radii of a circle are congruent, so two sides are congruent.

The answer is Definition of an isosceles triangle.

Where students go wrong

Forgetting that the radii are congruent without being told. It is a property of the circle, so it needs no Given, but it does need its own line.

Also called proofs with circles, chord proofs, inscribed angle proofs or radius perpendicular to a chord.

What you can put on this worksheet

Fill in the missing reasons
OMNPSQR
Given: MN‾ is tangent to circle O at N
  MP‾ is tangent to circle O at P
  QR‾ is tangent to circle O at S
Prove: QR=QN+RP
StatementsReasons1.  MN‾ is tangent to circle O at NGiven2.  MP‾ is tangent to circle O at PGiven3.  QR‾ is tangent to circle O at SGiven4.  QS=QNTangent segments from the same external point are congruent5.  SR=RP  6.  QS+SR=QRSegment addition postulate7.  QR=QN+RP  
→ Answer
5.  Tangent segments from the same external point are congruent
7.  Substitution
Fill in every reason
OMNPSQR
Given: MN‾ is tangent to circle O at N
  MP‾ is tangent to circle O at P
  QR‾ is tangent to circle O at S
Prove: QR=QN+RP
Use the word bank.
StatementsReasons1.  MN‾ is tangent to circle O at N  2.  MP‾ is tangent to circle O at P  3.  QR‾ is tangent to circle O at S  4.  QS=QN  5.  SR=RP  6.  QS+SR=QR  7.  QR=QN+RP  
Word bank: Parallel chords intercept congruent arcs · Given · Segment addition postulate · Definition of an angle bisector · Tangent segments from the same external point are congruent · Corresponding sides of similar triangles are proportional · Definition of a right angle · Substitution · If corresponding angles are congruent, the lines are parallel · Converse of the perpendicular bisector theorem
→ Answer
1.  Given
2.  Given
3.  Given
4.  Tangent segments from the same external point are congruent
5.  Tangent segments from the same external point are congruent
6.  Segment addition postulate
7.  Substitution
Fill in the missing statements
OMNPSQR
Given: MN‾ is tangent to circle O at N
  MP‾ is tangent to circle O at P
  QR‾ is tangent to circle O at S
Prove: QR=QN+RP
StatementsReasons1.  MN‾ is tangent to circle O at NGiven2.  MP‾ is tangent to circle O at PGiven3.  QR‾ is tangent to circle O at SGiven4.    Tangent segments from the same external point are congruent5.  SR=RPTangent segments from the same external point are congruent6.    Segment addition postulate7.  QR=QN+RPSubstitution
→ Answer
4.  QS=QN
6.  QS+SR=QR
Write the whole proof
OMNPSQR
Given: MN‾ is tangent to circle O at N
  MP‾ is tangent to circle O at P
  QR‾ is tangent to circle O at S
Prove: QR=QN+RP
→ Answer
One possible proof:
StatementsReasons1.  MN‾ is tangent to circle O at NGiven2.  MP‾ is tangent to circle O at PGiven3.  QR‾ is tangent to circle O at SGiven4.  QS=QNTangent segments from the same external point are congruent5.  SR=RPTangent segments from the same external point are congruent6.  QS+SR=QRSegment addition postulate7.  QR=QN+RPSubstitution
Name the postulate that finishes the proof
OKLMN
Given: K,L,M, and N lie on circle O
  KL‾≅MN‾
Prove: △KOL≅△MON
Which congruence postulate or theorem proves the triangles congruent?
StatementsReasons1.  K,L,M, and N lie on circle OGiven2.  KL‾≅MN‾Given3.  OK‾≅OM‾All radii of a circle are congruent4.  OL‾≅ON‾All radii of a circle are congruent5.  △KOL≅△MON  
→ Answer
SSS

Questions about these worksheets

Yes. Every circle 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 circle 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.

Forgetting that the radii are congruent without being told. It is a property of the circle, so it needs no Given, but it does need its own line.