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.
\egin{aligned}&\textbf{Given: } \ ext{circle } O \\ &\qquad\quad\; \overline{OC} \perp \overline{AB} at D \\&\textbf{Prove: } \overline{AD} \cong \overline{BD}\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \ ext{circle } O & \text{Given} \\\hline 2.\; \overline{OC} \perp \overline{AB} \text{ at } D & \text{Given} \\\hline 3.\; \angle ODA \text{ and } \angle ODB \text{ are right angles} & \; \\\hline 4.\; \overline{OA} \cong \overline{OB} & \text{All radii of a circle are congruent} \\\hline 5.\; \overline{OD} \cong \overline{OD} & \; \\\hline 6.\; \triangle ODA \cong \triangle ODB & \text{HL} \\\hline 7.\; \overline{AD} \cong \overline{BD} & \; \\\hline\end{array}3.Definition of perpendicular5.Reflexive property7.CPCTC
\egin{aligned}&\textbf{Given: } \ ext{circle } O \\ &\qquad\quad\; \overline{OT} \perp \overline{RS} at U \\&\textbf{Prove: } \overline{RU} \cong \overline{SU}\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \ ext{circle } O & \; \\\hline 2.\; \overline{OT} \perp \overline{RS} \text{ at } U & \; \\\hline 3.\; \angle OUR \text{ and } \angle OUS \text{ are right angles} & \; \\\hline 4.\; \overline{OR} \cong \overline{OS} & \; \\\hline 5.\; \overline{OU} \cong \overline{OU} & \; \\\hline 6.\; \triangle OUR \cong \triangle OUS & \; \\\hline 7.\; \overline{RU} \cong \overline{SU} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Supplements of congruent angles are congruent} \;\cdot\; \text{HL} \;\cdot\; \text{CPCTC} \;\cdot\; \text{All right angles are congruent} \;\cdot\; \text{Given} \;\cdot\; \text{All radii of a circle are congruent} \;\cdot\; \text{If base angles are congruent, the triangle is isosceles} \;\cdot\; \text{Complements of congruent angles are congruent} \;\cdot\; \text{Reflexive property} \;\cdot\; \text{Definition of perpendicular}1.Given2.Given3.Definition of perpendicular4.All radii of a circle are congruent5.Reflexive property6.HL7.CPCTC
\egin{aligned}&\textbf{Given: } \ ext{circle } O \\&\textbf{Prove: } \angle OAB \cong \angle OBA\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \ ext{circle } O & \text{Given} \\\hline 2.\; \overline{OA} \cong \overline{OB} & \; \\\hline 3.\; \triangle OAB \text{ is isosceles} & \; \\\hline 4.\; \angle OAB \cong \angle OBA & \; \\\hline\end{array}2.All radii of a circle are congruent3.Definition of an isosceles triangle4.Base angles of an isosceles triangle are congruent
\egin{aligned}&\textbf{Given: } \ ext{circle } O \\&\textbf{Prove: } \angle OFG \cong \angle OGF\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \ ext{circle } O & \; \\\hline 2.\; \overline{OF} \cong \overline{OG} & \; \\\hline 3.\; \triangle OFG \text{ is isosceles} & \; \\\hline 4.\; \angle OFG \cong \angle OGF & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Angle addition postulate} \;\cdot\; \text{All radii of a circle are congruent} \;\cdot\; \text{Given} \;\cdot\; \text{Corresponding angles of parallel lines are congruent} \;\cdot\; \text{Base angles of an isosceles triangle are congruent} \;\cdot\; \text{Definition of an isosceles triangle} \;\cdot\; \text{AA similarity}1.Given2.All radii of a circle are congruent3.Definition of an isosceles triangle4.Base angles of an isosceles triangle are congruent
\egin{aligned}&\textbf{Given: } \ ext{circle } O \\&\textbf{Prove: } \angle OJK \cong \angle OKJ\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \ ext{circle } O & \text{Given} \\\hline 2.\; \overline{OJ} \cong \overline{OK} & \; \\\hline 3.\; \triangle OJK \text{ is isosceles} & \; \\\hline 4.\; \angle OJK \cong \angle OKJ & \; \\\hline\end{array}2.All radii of a circle are congruent3.Definition of an isosceles triangle4.Base angles of an isosceles triangle are congruent
\egin{aligned}&\textbf{Given: } \ ext{circle } O \\ &\qquad\quad\; \overline{OC} \perp \overline{AB} at D \\&\textbf{Prove: } \overline{AD} \cong \overline{BD}\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \ ext{circle } O & \; \\\hline 2.\; \overline{OC} \perp \overline{AB} \text{ at } D & \; \\\hline 3.\; \angle ODA \text{ and } \angle ODB \text{ are right angles} & \; \\\hline 4.\; \overline{OA} \cong \overline{OB} & \; \\\hline 5.\; \overline{OD} \cong \overline{OD} & \; \\\hline 6.\; \triangle ODA \cong \triangle ODB & \; \\\hline 7.\; \overline{AD} \cong \overline{BD} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Given} \;\cdot\; \text{Substitution} \;\cdot\; \text{Definition of a median} \;\cdot\; \text{HL} \;\cdot\; \text{Reflexive property} \;\cdot\; \text{Definition of perpendicular} \;\cdot\; \text{If same-side interior angles are supplementary, the lines are parallel} \;\cdot\; \text{CPCTC} \;\cdot\; \text{SAS} \;\cdot\; \text{All radii of a circle are congruent}1.Given2.Given3.Definition of perpendicular4.All radii of a circle are congruent5.Reflexive property6.HL7.CPCTC
\egin{aligned}&\textbf{Given: } \ ext{circle } O \\ &\qquad\quad\; \overline{OT} \perp \overline{RS} at U \\&\textbf{Prove: } \overline{RU} \cong \overline{SU}\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \ ext{circle } O & \text{Given} \\\hline 2.\; \overline{OT} \perp \overline{RS} \text{ at } U & \; \\\hline 3.\; \angle OUR \text{ and } \angle OUS \text{ are right angles} & \; \\\hline 4.\; \overline{OR} \cong \overline{OS} & \text{All radii of a circle are congruent} \\\hline 5.\; \overline{OU} \cong \overline{OU} & \text{Reflexive property} \\\hline 6.\; \triangle OUR \cong \triangle OUS & \text{HL} \\\hline 7.\; \overline{RU} \cong \overline{SU} & \; \\\hline\end{array}2.Given3.Definition of perpendicular7.CPCTC
\egin{aligned}&\textbf{Given: } \ ext{circle } O \\ &\qquad\quad\; \overline{OY} \perp \overline{WX} at Z \\&\textbf{Prove: } \overline{WZ} \cong \overline{XZ}\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \ ext{circle } O & \; \\\hline 2.\; \overline{OY} \perp \overline{WX} \text{ at } Z & \; \\\hline 3.\; \angle OZW \text{ and } \angle OZX \text{ are right angles} & \; \\\hline 4.\; \overline{OW} \cong \overline{OX} & \; \\\hline 5.\; \overline{OZ} \cong \overline{OZ} & \; \\\hline 6.\; \triangle OZW \cong \triangle OZX & \; \\\hline 7.\; \overline{WZ} \cong \overline{XZ} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{CPCTC} \;\cdot\; \text{Reflexive property} \;\cdot\; \text{HL} \;\cdot\; \text{Same-side interior angles of parallel lines are supplementary} \;\cdot\; \text{Given} \;\cdot\; \text{All radii of a circle are congruent} \;\cdot\; \text{Angle addition postulate} \;\cdot\; \text{ASA} \;\cdot\; \text{Definition of perpendicular} \;\cdot\; \text{SAS}1.Given2.Given3.Definition of perpendicular4.All radii of a circle are congruent5.Reflexive property6.HL7.CPCTC
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
\egin{aligned}&\textbf{Given: } \ ext{circle } O \\&\textbf{Prove: } \overline{OW} \cong \overline{OX}\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \ ext{circle } O & \text{Given} \\\hline 2.\; \overline{OW} \cong \overline{OX} & \; \\\hline\end{array}→ 2.All radii of a circle are congruent
Fill in every reason
\egin{aligned}&\textbf{Given: } \ ext{circle } O \\&\textbf{Prove: } \overline{OW} \cong \overline{OX}\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \ ext{circle } O & \; \\\hline 2.\; \overline{OW} \cong \overline{OX} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{All radii of a circle are congruent} \;\cdot\; \text{Substitution} \;\cdot\; \text{Supplements of congruent angles are congruent} \;\cdot\; \text{Given} \;\cdot\; \text{CPCTC}→ 1.Given2.All radii of a circle are congruent
Fill in the missing statements
\egin{aligned}&\textbf{Given: } \ ext{circle } O \\&\textbf{Prove: } \overline{OW} \cong \overline{OX}\end{aligned} \\[4pt] \text{Give the missing statements.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \ ext{circle } O & \text{Given} \\\hline 2.\; \; & \text{All radii of a circle are congruent} \\\hline\end{array}→ 2.OW≅OX
Write the whole proof
\egin{aligned}&\textbf{Given: } \ ext{circle } O \\&\textbf{Prove: } \overline{OW} \cong \overline{OX}\end{aligned} \\[4pt] \text{Write the proof. It takes 2 lines.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \; & \; \\\hline 2.\; \; & \; \\\hline\end{array}→ 1.extcircleOGiven2.OW≅OXAll radii of a circle are congruent
Name the postulate that finishes the proof
\egin{aligned}&\textbf{Given: } \ ext{circle } O \\ &\qquad\quad\; \overline{OY} \perp \overline{WX} at Z \\&\textbf{Prove: } \overline{WZ} \cong \overline{XZ}\end{aligned} \\[4pt] \text{Which postulate proves the triangles congruent?} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \ ext{circle } O & \text{Given} \\\hline 2.\; \overline{OY} \perp \overline{WX} \text{ at } Z & \text{Given} \\\hline 3.\; \angle OZW \text{ and } \angle OZX \text{ are right angles} & \text{Definition of perpendicular} \\\hline 4.\; \overline{OW} \cong \overline{OX} & \text{All radii of a circle are congruent} \\\hline 5.\; \overline{OZ} \cong \overline{OZ} & \text{Reflexive property} \\\hline 6.\; \triangle OZW \cong \triangle OZX & \; \\\hline\end{array}→ HL
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.