These proofs run the parallel-line theorems backwards. Instead of being told the lines are parallel and concluding the angles are congruent, you are given the angles and must conclude the lines are parallel. Each theorem has a converse: congruent alternate interior angles prove the lines parallel, so do congruent corresponding angles, and so do supplementary same-side interior angles.
\egin{aligned}&\textbf{Given: } \angle 8 and ∠3 are supplementary \\&\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 8 \text{ and } \angle 3 \text{ are supplementary} & \text{Given} \\\hline 2.\; \angle 8 \text{ and } \angle 7 \text{ are supplementary} & \; \\\hline 3.\; \angle 7 \cong \angle 3 & \; \\\hline 4.\; \text{the two lines are parallel} & \; \\\hline\end{array}
\egin{aligned}&\textbf{Given: } \angle 3 and ∠5 are supplementary \\&\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 \text{ and } \angle 5 \text{ are supplementary} & \; \\\hline 2.\; \text{the two lines are parallel} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{All right angles are congruent} \;\cdot\; \text{Segment addition postulate} \;\cdot\; \text{Definition of perpendicular} \;\cdot\; \text{If same-side interior angles are supplementary, the lines are parallel} \;\cdot\; \text{Given}
\egin{aligned}&\textbf{Given: } \angle 2 and ∠5 are supplementary \\&\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 2 \text{ and } \angle 5 \text{ are supplementary} & \text{Given} \\\hline 2.\; \angle 2 \text{ and } \angle 1 \text{ are supplementary} & \; \\\hline 3.\; \angle 1 \cong \angle 5 & \; \\\hline 4.\; \text{the two lines are parallel} & \; \\\hline\end{array}
\egin{aligned}&\textbf{Given: } \angle 6 and ∠4 are supplementary \\&\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 6 \text{ and } \angle 4 \text{ are supplementary} & \; \\\hline 2.\; \text{the two lines are parallel} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Definition of midpoint} \;\cdot\; \text{If same-side interior angles are supplementary, the lines are parallel} \;\cdot\; \text{Given} \;\cdot\; \text{Linear pairs are supplementary} \;\cdot\; \text{CPCTC}
\egin{aligned}&\textbf{Given: } \angle 3 and ∠5 are supplementary \\&\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 \text{ and } \angle 5 \text{ are supplementary} & \text{Given} \\\hline 2.\; \text{the two lines are parallel} & \; \\\hline\end{array}
\egin{aligned}&\textbf{Given: } \angle 2 and ∠5 are supplementary \\&\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 2 \text{ and } \angle 5 \text{ are supplementary} & \; \\\hline 2.\; \angle 2 \text{ and } \angle 1 \text{ are supplementary} & \; \\\hline 3.\; \angle 1 \cong \angle 5 & \; \\\hline 4.\; \text{the two lines are parallel} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Given} \;\cdot\; \text{If corresponding angles are congruent, the lines are parallel} \;\cdot\; \text{If same-side interior angles are supplementary, the lines are parallel} \;\cdot\; \text{Linear pairs are supplementary} \;\cdot\; \text{Supplements of congruent angles are congruent} \;\cdot\; \text{ASA} \;\cdot\; \text{Definition of a segment bisector}
\egin{aligned}&\textbf{Given: } \angle 7 and ∠4 are supplementary \\&\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 7 \text{ and } \angle 4 \text{ are supplementary} & \text{Given} \\\hline 2.\; \angle 7 \text{ and } \angle 8 \text{ are supplementary} & \; \\\hline 3.\; \angle 8 \cong \angle 4 & \; \\\hline 4.\; \text{the two lines are parallel} & \; \\\hline\end{array}
\egin{aligned}&\textbf{Given: } \angle 7 and ∠1 are supplementary \\&\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 7 \text{ and } \angle 1 \text{ are supplementary} & \; \\\hline 2.\; \angle 7 \text{ and } \angle 5 \text{ are supplementary} & \; \\\hline 3.\; \angle 5 \cong \angle 1 & \; \\\hline 4.\; \text{the two lines are parallel} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{If base angles are congruent, the triangle is isosceles} \;\cdot\; \text{Subtraction property} \;\cdot\; \text{If corresponding angles are congruent, the lines are parallel} \;\cdot\; \text{Linear pairs are supplementary} \;\cdot\; \text{Angle addition postulate} \;\cdot\; \text{Given} \;\cdot\; \text{Supplements of congruent angles are congruent}
\egin{aligned}&\textbf{Given: } \angle 8 and ∠3 are supplementary \\&\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 8 \text{ and } \angle 3 \text{ are supplementary} & \text{Given} \\\hline 2.\; \angle 8 \text{ and } \angle 7 \text{ are supplementary} & \; \\\hline 3.\; \angle 7 \cong \angle 3 & \; \\\hline 4.\; \text{the two lines are parallel} & \; \\\hline\end{array}2.Linear pairs are supplementary3.Supplements of congruent angles are congruent4.If corresponding angles are congruent, the lines are parallel
\egin{aligned}&\textbf{Given: } \angle 3 and ∠5 are supplementary \\&\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 \text{ and } \angle 5 \text{ are supplementary} & \; \\\hline 2.\; \text{the two lines are parallel} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{All right angles are congruent} \;\cdot\; \text{Segment addition postulate} \;\cdot\; \text{Definition of perpendicular} \;\cdot\; \text{If same-side interior angles are supplementary, the lines are parallel} \;\cdot\; \text{Given}1.Given2.If same-side interior angles are supplementary, the lines are parallel
\egin{aligned}&\textbf{Given: } \angle 2 and ∠5 are supplementary \\&\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 2 \text{ and } \angle 5 \text{ are supplementary} & \text{Given} \\\hline 2.\; \angle 2 \text{ and } \angle 1 \text{ are supplementary} & \; \\\hline 3.\; \angle 1 \cong \angle 5 & \; \\\hline 4.\; \text{the two lines are parallel} & \; \\\hline\end{array}2.Linear pairs are supplementary3.Supplements of congruent angles are congruent4.If corresponding angles are congruent, the lines are parallel
\egin{aligned}&\textbf{Given: } \angle 6 and ∠4 are supplementary \\&\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 6 \text{ and } \angle 4 \text{ are supplementary} & \; \\\hline 2.\; \text{the two lines are parallel} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Definition of midpoint} \;\cdot\; \text{If same-side interior angles are supplementary, the lines are parallel} \;\cdot\; \text{Given} \;\cdot\; \text{Linear pairs are supplementary} \;\cdot\; \text{CPCTC}1.Given2.If same-side interior angles are supplementary, the lines are parallel
\egin{aligned}&\textbf{Given: } \angle 3 and ∠5 are supplementary \\&\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 \text{ and } \angle 5 \text{ are supplementary} & \text{Given} \\\hline 2.\; \text{the two lines are parallel} & \; \\\hline\end{array}2.If same-side interior angles are supplementary, the lines are parallel
\egin{aligned}&\textbf{Given: } \angle 2 and ∠5 are supplementary \\&\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 2 \text{ and } \angle 5 \text{ are supplementary} & \; \\\hline 2.\; \angle 2 \text{ and } \angle 1 \text{ are supplementary} & \; \\\hline 3.\; \angle 1 \cong \angle 5 & \; \\\hline 4.\; \text{the two lines are parallel} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Given} \;\cdot\; \text{If corresponding angles are congruent, the lines are parallel} \;\cdot\; \text{If same-side interior angles are supplementary, the lines are parallel} \;\cdot\; \text{Linear pairs are supplementary} \;\cdot\; \text{Supplements of congruent angles are congruent} \;\cdot\; \text{ASA} \;\cdot\; \text{Definition of a segment bisector}1.Given2.Linear pairs are supplementary3.Supplements of congruent angles are congruent4.If corresponding angles are congruent, the lines are parallel
\egin{aligned}&\textbf{Given: } \angle 7 and ∠4 are supplementary \\&\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 7 \text{ and } \angle 4 \text{ are supplementary} & \text{Given} \\\hline 2.\; \angle 7 \text{ and } \angle 8 \text{ are supplementary} & \; \\\hline 3.\; \angle 8 \cong \angle 4 & \; \\\hline 4.\; \text{the two lines are parallel} & \; \\\hline\end{array}2.Linear pairs are supplementary3.Supplements of congruent angles are congruent4.If corresponding angles are congruent, the lines are parallel
\egin{aligned}&\textbf{Given: } \angle 7 and ∠1 are supplementary \\&\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 7 \text{ and } \angle 1 \text{ are supplementary} & \; \\\hline 2.\; \angle 7 \text{ and } \angle 5 \text{ are supplementary} & \; \\\hline 3.\; \angle 5 \cong \angle 1 & \; \\\hline 4.\; \text{the two lines are parallel} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{If base angles are congruent, the triangle is isosceles} \;\cdot\; \text{Subtraction property} \;\cdot\; \text{If corresponding angles are congruent, the lines are parallel} \;\cdot\; \text{Linear pairs are supplementary} \;\cdot\; \text{Angle addition postulate} \;\cdot\; \text{Given} \;\cdot\; \text{Supplements of congruent angles are congruent}1.Given2.Linear pairs are supplementary3.Supplements of congruent angles are congruent4.If corresponding angles are congruent, the lines are parallel
The answer is If alternate interior angles are congruent, the lines are parallel.
Where students go wrong
Quoting the forward theorem when you need the converse. "Alternate interior angles are congruent" assumes the lines are parallel, which is the thing you are trying to prove.
Also called converse of alternate interior angles, proving parallel lines, parallel line proofs or transversal proofs.
What you can put on this worksheet
Fill in the missing reasons
\egin{aligned}&\textbf{Given: } \angle 2 \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 2 \cong \angle 7 & \text{Given} \\\hline 2.\; \text{the two lines are parallel} & \; \\\hline\end{array}→ 2.If alternate exterior angles are congruent, the lines are parallel
Fill in every reason
\egin{aligned}&\textbf{Given: } \angle 2 \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 2 \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 alternate exterior angles are congruent, the lines are parallel} \;\cdot\; \text{Definition of a segment bisector}→ 1.Given2.If alternate exterior angles are congruent, the lines are parallel
Fill in the missing statements
\egin{aligned}&\textbf{Given: } \angle 2 \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 2 \cong \angle 7 & \text{Given} \\\hline 2.\; \; & \text{If alternate exterior 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 2 \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.∠2≅∠7Given2.the two lines are parallelIf alternate exterior angles are congruent, the lines are parallel
Questions about these worksheets
Yes. Every proving lines parallel 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 lines parallel 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.
Quoting the forward theorem when you need the converse. "Alternate interior angles are congruent" assumes the lines are parallel, which is the thing you are trying to prove.