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.
Select your difficulty Pick more than one for a sheet that mixes them.
Easy \egin{aligned}&\textbf{Given: } \overline{KL} \cong \overline{MN} \\&\textbf{Prove: } \overline{KM} \cong \overline{LN}\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overline{KL} \cong \overline{MN} & \text{Given} \\\hline 2.\; \overline{LM} \cong \overline{LM} & \; \\\hline 3.\; KL + LM = KM \ ext{ and } LM + MN = LN & \text{Segment addition postulate} \\\hline 4.\; \overline{KM} \cong \overline{LN} & \; \\\hline\end{array} 2. Reflexive property 4. Addition property 2. Reflexive property 4. Addition property Medium \egin{aligned}&\textbf{Given: } the two lines are parallel \\&\textbf{Prove: } \angle 6 and ∠ 4 ∠4 \end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \text{the two lines are parallel} & \text{Given} \\\hline 2.\; \angle 6 \cong \angle 3 & \; \\\hline 3.\; \angle 3 \text{ and } \angle 4 \text{ are supplementary} & \; \\\hline 4.\; \angle 6 \text{ and } \angle 4 \text{ are supplementary} & \; \\\hline\end{array} 2. Alternate interior angles of parallel lines are congruent 3. Linear pairs are supplementary 4. Substitution 2. Alternate interior angles of parallel lines are congruent 3. Linear pairs are supplementary 4. Substitution Hard \egin{aligned}&\textbf{Given: } \overrightarrow{VX} bisects \angle WVY \\ &\qquad\quad\; \angle YVZ \cong \angle WVX \\&\textbf{Prove: } \angle WVY \cong \angle XVZ\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overrightarrow{VX} \text{ bisects } \angle WVY & \text{Given} \\\hline 2.\; \angle WVX \cong \angle XVY & \text{Definition of an angle bisector} \\\hline 3.\; \angle YVZ \cong \angle WVX & \text{Given} \\\hline 4.\; \angle XVY \cong \angle YVZ & \; \\\hline 5.\; \angle WVX + \angle XVY = \angle WVY \text{ and } \angle XVY + \angle YVZ = \angle XVZ & \; \\\hline 6.\; \angle WVY \cong \angle XVZ & \; \\\hline\end{array} 4. Transitive property 5. Angle addition postulate 6. Addition property 4. Transitive property 5. Angle addition postulate 6. Addition property
Question order
Mixed One kind at a time
Printing options Working space
Roomy Compact
Generate Download PDF Print instead Free, no account. Make as many different versions as you need.
Sigma Prep · sigmaprep.io/worksheets
Name
Segment and Angle Proofs Date
Complete each proof.
1 2 3 4 5 6 7 8 \egin{aligned}&\textbf{Given: } the two lines are parallel \\&\textbf{Prove: } \angle 6 and ∠ 4 ∠4 \end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \text{the two lines are parallel} & \text{Given} \\\hline 2.\; \angle 6 \cong \angle 3 & \; \\\hline 3.\; \angle 3 \text{ and } \angle 4 \text{ are supplementary} & \; \\\hline 4.\; \angle 6 \text{ and } \angle 4 \text{ are supplementary} & \; \\\hline\end{array} 1 2 3 4 5 6 7 8 \egin{aligned}&\textbf{Given: } the two lines are parallel \\&\textbf{Prove: } \angle 4 and ∠ 6 ∠6 \end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \text{the two lines are parallel} & \; \\\hline 2.\; \angle 4 \cong \angle 5 & \; \\\hline 3.\; \angle 5 \text{ and } \angle 6 \text{ are supplementary} & \; \\\hline 4.\; \angle 4 \text{ and } \angle 6 \text{ are supplementary} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Substitution} \;\cdot\; \text{Base angles of an isosceles triangle are congruent} \;\cdot\; \text{All right angles are congruent} \;\cdot\; \text{Angle addition postulate} \;\cdot\; \text{Linear pairs are supplementary} \;\cdot\; \text{Given} \;\cdot\; \text{Alternate interior angles of parallel lines are congruent} P Q R S \egin{aligned}&\textbf{Given: } Q is the midpoint of \overline{PR} \\ &\qquad\quad\; \overline{PQ} \cong \overline{RS} \\&\textbf{Prove: } \overline{QR} \cong \overline{RS}\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; Q \text{ is the midpoint of } \overline{PR} & \text{Given} \\\hline 2.\; \overline{PQ} \cong \overline{QR} & \; \\\hline 3.\; \overline{PQ} \cong \overline{RS} & \; \\\hline 4.\; \overline{QR} \cong \overline{RS} & \; \\\hline\end{array} P Q R S \egin{aligned}&\textbf{Given: } Q is the midpoint of \overline{PR} \\ &\qquad\quad\; \overline{PQ} \cong \overline{RS} \\&\textbf{Prove: } \overline{QR} \cong \overline{RS}\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; Q \text{ is the midpoint of } \overline{PR} & \; \\\hline 2.\; \overline{PQ} \cong \overline{QR} & \; \\\hline 3.\; \overline{PQ} \cong \overline{RS} & \; \\\hline 4.\; \overline{QR} \cong \overline{RS} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Definition of a segment bisector} \;\cdot\; \text{Definition of midpoint} \;\cdot\; \text{Given} \;\cdot\; \text{Corresponding angles of parallel lines are congruent} \;\cdot\; \text{HL} \;\cdot\; \text{Transitive property} \;\cdot\; \text{AA similarity} J K L M V \egin{aligned}&\textbf{Given: } \overrightarrow{VK} bisects \angle JVL \\ &\qquad\quad\; \angle JVK \cong \angle LVM \\&\textbf{Prove: } \angle KVL \cong \angle LVM\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overrightarrow{VK} \text{ bisects } \angle JVL & \text{Given} \\\hline 2.\; \angle JVK \cong \angle KVL & \; \\\hline 3.\; \angle JVK \cong \angle LVM & \; \\\hline 4.\; \angle KVL \cong \angle LVM & \; \\\hline\end{array} A B C D \egin{aligned}&\textbf{Given: } \overline{AC} \cong \overline{BD} \\&\textbf{Prove: } \overline{AB} \cong \overline{CD}\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overline{AC} \cong \overline{BD} & \; \\\hline 2.\; AB + BC = AC \ ext{ and } BC + CD = BD & \; \\\hline 3.\; AB + BC = BC + CD & \; \\\hline 4.\; \overline{AB} \cong \overline{CD} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Subtraction property} \;\cdot\; \text{Given} \;\cdot\; \text{Transitive property} \;\cdot\; \text{Substitution} \;\cdot\; \text{Definition of an altitude} \;\cdot\; \text{Segment addition postulate} \;\cdot\; \text{AAS} 1 2 3 4 5 6 7 8 \egin{aligned}&\textbf{Given: } \angle 8 \cong \angle 1 \\&\textbf{Prove: } \angle 5 \cong \angle 1\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 \cong \angle 1 & \text{Given} \\\hline 2.\; \angle 8 \cong \angle 5 & \; \\\hline 3.\; \angle 5 \cong \angle 1 & \; \\\hline\end{array} 1 2 3 4 5 6 7 8 \egin{aligned}&\textbf{Given: } \angle 3 \cong \angle 6 \\&\textbf{Prove: } \angle 2 \cong \angle 6\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \angle 3 \cong \angle 6 & \; \\\hline 2.\; \angle 3 \cong \angle 2 & \; \\\hline 3.\; \angle 2 \cong \angle 6 & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{HL} \;\cdot\; \text{Vertical angles are congruent} \;\cdot\; \text{Definition of an isosceles triangle} \;\cdot\; \text{Given} \;\cdot\; \text{Transitive property} \;\cdot\; \text{Linear pairs are supplementary}
Sigma Prep · sigmaprep.io/worksheets
Name
Segment and Angle ProofsAnswer keyVersion 1 Date
Complete each proof.
1 2 3 4 5 6 7 8 \egin{aligned}&\textbf{Given: } the two lines are parallel \\&\textbf{Prove: } \angle 6 and ∠ 4 ∠4 \end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \text{the two lines are parallel} & \text{Given} \\\hline 2.\; \angle 6 \cong \angle 3 & \; \\\hline 3.\; \angle 3 \text{ and } \angle 4 \text{ are supplementary} & \; \\\hline 4.\; \angle 6 \text{ and } \angle 4 \text{ are supplementary} & \; \\\hline\end{array} 2. Alternate interior angles of parallel lines are congruent 3. Linear pairs are supplementary 4. Substitution 2. Alternate interior angles of parallel lines are congruent 3. Linear pairs are supplementary 4. Substitution 1 2 3 4 5 6 7 8 \egin{aligned}&\textbf{Given: } the two lines are parallel \\&\textbf{Prove: } \angle 4 and ∠ 6 ∠6 \end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \text{the two lines are parallel} & \; \\\hline 2.\; \angle 4 \cong \angle 5 & \; \\\hline 3.\; \angle 5 \text{ and } \angle 6 \text{ are supplementary} & \; \\\hline 4.\; \angle 4 \text{ and } \angle 6 \text{ are supplementary} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Substitution} \;\cdot\; \text{Base angles of an isosceles triangle are congruent} \;\cdot\; \text{All right angles are congruent} \;\cdot\; \text{Angle addition postulate} \;\cdot\; \text{Linear pairs are supplementary} \;\cdot\; \text{Given} \;\cdot\; \text{Alternate interior angles of parallel lines are congruent} 1. Given 2. Alternate interior angles of parallel lines are congruent 3. Linear pairs are supplementary 4. Substitution 1. Given 2. Alternate interior angles of parallel lines are congruent 3. Linear pairs are supplementary 4. Substitution P Q R S \egin{aligned}&\textbf{Given: } Q is the midpoint of \overline{PR} \\ &\qquad\quad\; \overline{PQ} \cong \overline{RS} \\&\textbf{Prove: } \overline{QR} \cong \overline{RS}\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; Q \text{ is the midpoint of } \overline{PR} & \text{Given} \\\hline 2.\; \overline{PQ} \cong \overline{QR} & \; \\\hline 3.\; \overline{PQ} \cong \overline{RS} & \; \\\hline 4.\; \overline{QR} \cong \overline{RS} & \; \\\hline\end{array} 2. Definition of midpoint 3. Given 4. Transitive property 2. Definition of midpoint 3. Given 4. Transitive property P Q R S \egin{aligned}&\textbf{Given: } Q is the midpoint of \overline{PR} \\ &\qquad\quad\; \overline{PQ} \cong \overline{RS} \\&\textbf{Prove: } \overline{QR} \cong \overline{RS}\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; Q \text{ is the midpoint of } \overline{PR} & \; \\\hline 2.\; \overline{PQ} \cong \overline{QR} & \; \\\hline 3.\; \overline{PQ} \cong \overline{RS} & \; \\\hline 4.\; \overline{QR} \cong \overline{RS} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Definition of a segment bisector} \;\cdot\; \text{Definition of midpoint} \;\cdot\; \text{Given} \;\cdot\; \text{Corresponding angles of parallel lines are congruent} \;\cdot\; \text{HL} \;\cdot\; \text{Transitive property} \;\cdot\; \text{AA similarity} 1. Given 2. Definition of midpoint 3. Given 4. Transitive property 1. Given 2. Definition of midpoint 3. Given 4. Transitive property J K L M V \egin{aligned}&\textbf{Given: } \overrightarrow{VK} bisects \angle JVL \\ &\qquad\quad\; \angle JVK \cong \angle LVM \\&\textbf{Prove: } \angle KVL \cong \angle LVM\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overrightarrow{VK} \text{ bisects } \angle JVL & \text{Given} \\\hline 2.\; \angle JVK \cong \angle KVL & \; \\\hline 3.\; \angle JVK \cong \angle LVM & \; \\\hline 4.\; \angle KVL \cong \angle LVM & \; \\\hline\end{array} 2. Definition of an angle bisector 3. Given 4. Transitive property 2. Definition of an angle bisector 3. Given 4. Transitive property A B C D \egin{aligned}&\textbf{Given: } \overline{AC} \cong \overline{BD} \\&\textbf{Prove: } \overline{AB} \cong \overline{CD}\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overline{AC} \cong \overline{BD} & \; \\\hline 2.\; AB + BC = AC \ ext{ and } BC + CD = BD & \; \\\hline 3.\; AB + BC = BC + CD & \; \\\hline 4.\; \overline{AB} \cong \overline{CD} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Subtraction property} \;\cdot\; \text{Given} \;\cdot\; \text{Transitive property} \;\cdot\; \text{Substitution} \;\cdot\; \text{Definition of an altitude} \;\cdot\; \text{Segment addition postulate} \;\cdot\; \text{AAS} 1. Given 2. Segment addition postulate 3. Substitution 4. Subtraction property 1. Given 2. Segment addition postulate 3. Substitution 4. Subtraction property 1 2 3 4 5 6 7 8 \egin{aligned}&\textbf{Given: } \angle 8 \cong \angle 1 \\&\textbf{Prove: } \angle 5 \cong \angle 1\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 \cong \angle 1 & \text{Given} \\\hline 2.\; \angle 8 \cong \angle 5 & \; \\\hline 3.\; \angle 5 \cong \angle 1 & \; \\\hline\end{array} 2. Vertical angles are congruent 3. Transitive property 2. Vertical angles are congruent 3. Transitive property 1 2 3 4 5 6 7 8 \egin{aligned}&\textbf{Given: } \angle 3 \cong \angle 6 \\&\textbf{Prove: } \angle 2 \cong \angle 6\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \angle 3 \cong \angle 6 & \; \\\hline 2.\; \angle 3 \cong \angle 2 & \; \\\hline 3.\; \angle 2 \cong \angle 6 & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{HL} \;\cdot\; \text{Vertical angles are congruent} \;\cdot\; \text{Definition of an isosceles triangle} \;\cdot\; \text{Given} \;\cdot\; \text{Transitive property} \;\cdot\; \text{Linear pairs are supplementary} 1. Given 2. Vertical angles are congruent 3. Transitive property 1. Given 2. Vertical angles are congruent 3. Transitive property
Found a mistake on this sheet?
How to do these Given A B ‾ ≅ C D ‾ A B ≅ C D A C ‾ ≅ B D ‾ A C ≅ B D
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 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 \egin{aligned}&\textbf{Given: } the two lines are parallel \\&\textbf{Prove: } \angle 4 \cong \angle 5\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \text{the two lines are parallel} & \text{Given} \\\hline 2.\; \angle 4 \cong \angle 5 & \; \\\hline\end{array} → 2. Alternate interior angles of parallel lines are congruent 2. Alternate interior angles of parallel lines are congruent
Fill in every reason \egin{aligned}&\textbf{Given: } the two lines are parallel \\&\textbf{Prove: } \angle 4 \cong \angle 5\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \text{the two lines are parallel} & \; \\\hline 2.\; \angle 4 \cong \angle 5 & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Given} \;\cdot\; \text{SSS} \;\cdot\; \text{Supplements of congruent angles are congruent} \;\cdot\; \text{Alternate interior angles of parallel lines are congruent} \;\cdot\; \text{Definition of a segment bisector} → 1. Given 2. Alternate interior angles of parallel lines are congruent 1. Given 2. Alternate interior angles of parallel lines are congruent
Fill in the missing statements \egin{aligned}&\textbf{Given: } the two lines are parallel \\&\textbf{Prove: } \angle 4 \cong \angle 5\end{aligned} \\[4pt] \text{Give the missing statements.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \text{the two lines are parallel} & \text{Given} \\\hline 2.\; \; & \text{Alternate interior angles of parallel lines are congruent} \\\hline\end{array} → 2. ∠ 4 ≅ ∠ 5 2. ∠4 ≅ ∠5
Write the whole proof \egin{aligned}&\textbf{Given: } the two lines are parallel \\&\textbf{Prove: } \angle 4 \cong \angle 5\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. the two lines are parallel Given 2. ∠ 4 ≅ ∠ 5 Alternate interior angles of parallel lines are congruent 1. the two lines are parallel Given 2. ∠4 ≅ ∠5 Alternate interior angles of parallel lines are congruent Questions about these worksheets Are these segment and angle proofs worksheets free? + 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.
Do the segment and angle proofs worksheets come with an answer key? + 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.
Are these segment and angle proofs worksheets printable? + 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.
What grade are these segment and angle proofs worksheets for? + 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.
Can I get a different version of the same segment and angle proofs worksheet? + 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.
Is this a free alternative to Kuta Software for segment and angle proofs? + 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.
Can I choose how hard the segment and angle proofs questions are? + 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.
What kinds of question are on a segment and angle proofs worksheet? + 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.
What do students get wrong with segment and angle proofs? + Skipping the addition postulate line. Going straight from the Given to the conclusion leaves out the step that actually justifies it.
Proofs All geometry worksheets (91) Pre-Algebra 70 Algebra 1 121 Algebra 2 115 Precalculus 115 Calculus 59 Statistics 19