A two-column proof is an argument written twice: what you know on the left, why you know it on the right. Every line comes from the Given, from a definition, from a property like the reflexive property, or from a theorem you have already proved. Nothing comes from how the picture looks. Once two triangles are congruent, CPCTC lets you carry any matching pair of sides or angles across, which is how most proofs finish.
BC is congruent to itself by the reflexive property.
Segment addition turns AB + BC into AC, and BC + CD into BD.
Adding the same segment to congruent segments keeps them congruent.
The answer is Addition property.
Where students go wrong
Using something the picture shows but the Given never says. If two segments look equal and no mark or Given says so, you cannot use it.
Also called geometry proofs, triangle congruence proofs, statements and reasons, fill in the blank proofs or CPCTC proofs.
What you can put on this worksheet
Fill in the missing reasons
\egin{aligned}&\textbf{Given: } \angle 3 \cong \angle 7 \\&\textbf{Prove: } the two lines are parallel\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \angle 3 \cong \angle 7 & \text{Given} \\\hline 2.\; \text{the two lines are parallel} & \; \\\hline\end{array}→ 2.If corresponding angles are congruent, the lines are parallel
Fill in every reason
\egin{aligned}&\textbf{Given: } \angle 3 \cong \angle 7 \\&\textbf{Prove: } the two lines are parallel\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \angle 3 \cong \angle 7 & \; \\\hline 2.\; \text{the two lines are parallel} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Given} \;\cdot\; \text{Corresponding angles of parallel lines are congruent} \;\cdot\; \text{Supplements of congruent angles are congruent} \;\cdot\; \text{If corresponding angles are congruent, the lines are parallel} \;\cdot\; \text{Definition of a segment bisector}→ 1.Given2.If corresponding angles are congruent, the lines are parallel
Fill in the missing statements
\egin{aligned}&\textbf{Given: } \angle 3 \cong \angle 7 \\&\textbf{Prove: } the two lines are parallel\end{aligned} \\[4pt] \text{Give the missing statements.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \angle 3 \cong \angle 7 & \text{Given} \\\hline 2.\; \; & \text{If corresponding angles are congruent, the lines are parallel} \\\hline\end{array}→ 2.the two lines are parallel
Write the whole proof
\egin{aligned}&\textbf{Given: } \angle 3 \cong \angle 7 \\&\textbf{Prove: } the two lines are parallel\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.∠3≅∠7Given2.the two lines are parallelIf corresponding angles are congruent, the lines are parallel
Name the postulate that finishes the proof
\egin{aligned}&\textbf{Given: } \overline{WX} \cong \overline{YX} \\ &\qquad\quad\; \overline{XZ} is a median of \overline{WY} \\&\textbf{Prove: } \triangle WXZ \cong \triangle YXZ\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.\; \overline{WX} \cong \overline{YX} & \text{Given} \\\hline 2.\; \overline{XZ} \text{ is a median of } \overline{WY} & \text{Given} \\\hline 3.\; \overline{WZ} \cong \overline{YZ} & \text{Definition of a median} \\\hline 4.\; \overline{XZ} \cong \overline{XZ} & \text{Reflexive property} \\\hline 5.\; \triangle WXZ \cong \triangle YXZ & \; \\\hline\end{array}→ SSS
Questions about these worksheets
Yes. Every two-column 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 two-column 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 something the picture shows but the Given never says. If two segments look equal and no mark or Given says so, you cannot use it.