eginalignedGiven: AB is congruent to CB; BD bisects ∠ ABC; Prove: ∠ A is congruent to ∠ C; [4pt] Give the missing reasons.; [2pt] 1.7eginarray|l|l| extbfStatements extbfReasons; 1. AB is congruent to CB Given; 2. BD bisects ∠ ABC Given; 3. ∠ ABD is congruent to ∠ CBD; 4. BD is congruent to BD; 5. triangle ABD is congruent to triangle CBD SAS; 6. ∠ A is congruent to ∠ C;

3. Definition of an angle bisector; 4. Reflexive property; 6. CPCTC

eginalignedGiven: RS is congruent to TS; SU bisects ∠ RST; Prove: ∠ R is congruent to ∠ T; [4pt] 1.7eginarray|l|l| extbfStatements extbfReasons; 1. RS is congruent to TS; 2. SU bisects ∠ RST; 3. ∠ RSU is congruent to ∠ TSU; 4. SU is congruent to SU; 5. triangle RSU is congruent to triangle TSU; 6. ∠ R is congruent to ∠ T;; [3pt] Word bank: Supplements of congruent angles are congruent × CPCTC × If base angles are congruent, the triangle is isosceles × Linear pairs are supplementary × Reflexive property × Given × Base angles of an isosceles triangle are congruent × SAS × Definition of an angle bisector

1. Given; 2. Given; 3. Definition of an angle bisector; 4. Reflexive property; 5. SAS; 6. CPCTC

eginalignedGiven: T is the midpoint of PS and QR; Prove: PQ is congruent to SR; [4pt] Give the missing reasons.; [2pt] 1.7eginarray|l|l| extbfStatements extbfReasons; 1. T is the midpoint of PS and QR Given; 2. PT is congruent to ST; 3. QT is congruent to RT Definition of midpoint; 4. ∠ PTQ is congruent to ∠ STR; 5. triangle PQT is congruent to triangle SRT; 6. PQ is congruent to SR CPCTC;

2. Definition of midpoint; 4. Vertical angles are congruent; 5. SAS

eginalignedGiven: T is the midpoint of PS and QR; Prove: PQ is congruent to SR; [4pt] 1.7eginarray|l|l| extbfStatements extbfReasons; 1. T is the midpoint of PS and QR; 2. PT is congruent to ST; 3. QT is congruent to RT; 4. ∠ PTQ is congruent to ∠ STR; 5. triangle PQT is congruent to triangle SRT; 6. PQ is congruent to SR;; [3pt] Word bank: CPCTC × Vertical angles are congruent × Given × AA similarity × Definition of midpoint × Angle addition postulate × AAS × SAS × Alternate interior angles of parallel lines are congruent

1. Given; 2. Definition of midpoint; 3. Definition of midpoint; 4. Vertical angles are congruent; 5. SAS; 6. CPCTC

eginalignedGiven: T is the midpoint of PS and QR; Prove: PQ is congruent to SR; [4pt] Give the missing reasons.; [2pt] 1.7eginarray|l|l| extbfStatements extbfReasons; 1. T is the midpoint of PS and QR Given; 2. PT is congruent to ST; 3. QT is congruent to RT Definition of midpoint; 4. ∠ PTQ is congruent to ∠ STR Vertical angles are congruent; 5. triangle PQT is congruent to triangle SRT; 6. PQ is congruent to SR;

2. Definition of midpoint; 5. SAS; 6. CPCTC

eginalignedGiven: P is the midpoint of KN and LM; Prove: KL is congruent to NM; [4pt] 1.7eginarray|l|l| extbfStatements extbfReasons; 1. P is the midpoint of KN and LM; 2. KP is congruent to NP; 3. LP is congruent to MP; 4. ∠ KPL is congruent to ∠ NPM; 5. triangle KLP is congruent to triangle NMP; 6. KL is congruent to NM;; [3pt] Word bank: Definition of midpoint × Vertical angles are congruent × Given × SAS × All right angles are congruent × All radii of a circle are congruent × CPCTC × Definition of a median × HL

1. Given; 2. Definition of midpoint; 3. Definition of midpoint; 4. Vertical angles are congruent; 5. SAS; 6. CPCTC

eginalignedGiven: JK is congruent to LK; KM bisects ∠ JKL; Prove: ∠ J is congruent to ∠ L; [4pt] Give the missing reasons.; [2pt] 1.7eginarray|l|l| extbfStatements extbfReasons; 1. JK is congruent to LK Given; 2. KM bisects ∠ JKL Given; 3. ∠ JKM is congruent to ∠ LKM Definition of an angle bisector; 4. KM is congruent to KM; 5. triangle JKM is congruent to triangle LKM; 6. ∠ J is congruent to ∠ L;

4. Reflexive property; 5. SAS; 6. CPCTC

eginalignedGiven: J is the midpoint of FI and GH; Prove: FG is congruent to IH; [4pt] 1.7eginarray|l|l| extbfStatements extbfReasons; 1. J is the midpoint of FI and GH; 2. FJ is congruent to IJ; 3. GJ is congruent to HJ; 4. ∠ FJG is congruent to ∠ IJH; 5. triangle FGJ is congruent to triangle IHJ; 6. FG is congruent to IH;; [3pt] Word bank: CPCTC × If base angles are congruent, the triangle is isosceles × If both pairs of opposite sides are parallel, the figure is a parallelogram × Subtraction property × Angle addition postulate × Definition of midpoint × Vertical angles are congruent × Given × SAS

1. Given; 2. Definition of midpoint; 3. Definition of midpoint; 4. Vertical angles are congruent; 5. SAS; 6. CPCTC

eginalignedGiven: WX is congruent to YZ; XY is congruent to ZW; Prove: ∠ X is congruent to ∠ Z; [4pt] 1.7eginarray|l|l| extbfStatements extbfReasons; 1. WX is congruent to YZ; 2. XY is congruent to ZW; 3. WY is congruent to WY; 4. triangle WXY is congruent to triangle YZW; 5. ∠ X is congruent to ∠ Z;; [3pt] Word bank: Corresponding angles of parallel lines are congruent × Given × CPCTC × Addition property × Inscribed angles that intercept the same arc are congruent × SSS × Supplements of congruent angles are congruent × Reflexive property

1. Given; 2. Given; 3. Reflexive property; 4. SSS; 5. CPCTC

eginalignedGiven: WX is congruent to YZ; XY is congruent to ZW; Prove: ∠ X is congruent to ∠ Z; [4pt] Give the missing statements.; [2pt] 1.7eginarray|l|l| extbfStatements extbfReasons; 1. WX is congruent to YZ Given; 2. Given; 3. WY is congruent to WY Reflexive property; 4. SSS; 5. ∠ X is congruent to ∠ Z CPCTC;

2. XY is congruent to ZW; 4. triangle WXY is congruent to triangle YZW

eginalignedGiven: WX is congruent to YZ; XY is congruent to ZW; Prove: ∠ X is congruent to ∠ Z; [4pt] Write the proof. It takes 5 lines.; [2pt] 1.7eginarray|l|l| extbfStatements extbfReasons; 1.; 2.; 3.; 4.; 5.;

1. WX is congruent to YZ; Given; 2. XY is congruent to ZW; Given; 3. WY is congruent to WY; Reflexive property; 4. triangle WXY is congruent to triangle YZW; SSS; 5. ∠ X is congruent to ∠ Z; CPCTC

Worksheets · Geometry

CPCTC proofs worksheet

CPCTC stands for corresponding parts of congruent triangles are congruent, and it is always the last move, never the first. Prove the two triangles congruent by SSS, SAS, ASA, AAS or HL, and only then may you say that any other matching pair of sides or angles is congruent. Writing CPCTC before the triangles are congruent is the one thing these sheets are built to catch.

Select your difficulty

Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name CPCTC Proofs Date Period Complete each proof. $\egin{aligned}&\textbf{Given: } \overline{AB} \cong \overline{CB} \\ &\qquad\quad\; \overline{BD}$ bisects $\angle ABC \\&\textbf{Prove: } \angle A \cong \angle C\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{\|l\|l\|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overline{AB} \cong \overline{CB} & \text{Given} \\\hline 2.\; \overline{BD} \text{ bisects } \angle ABC & \text{Given} \\\hline 3.\; \angle ABD \cong \angle CBD & \; \\\hline 4.\; \overline{BD} \cong \overline{BD} & \; \\\hline 5.\; \triangle ABD \cong \triangle CBD & \text{SAS} \\\hline 6.\; \angle A \cong \angle C & \; \\\hline\end{array}$ $\egin{aligned}&\textbf{Given: } \overline{RS} \cong \overline{TS} \\ &\qquad\quad\; \overline{SU}$ bisects $\angle RST \\&\textbf{Prove: } \angle R \cong \angle T\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{\|l\|l\|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overline{RS} \cong \overline{TS} & \; \\\hline 2.\; \overline{SU} \text{ bisects } \angle RST & \; \\\hline 3.\; \angle RSU \cong \angle TSU & \; \\\hline 4.\; \overline{SU} \cong \overline{SU} & \; \\\hline 5.\; \triangle RSU \cong \triangle TSU & \; \\\hline 6.\; \angle R \cong \angle T & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Supplements of congruent angles are congruent} \;\cdot\; \text{CPCTC} \;\cdot\; \text{If base angles are congruent, the triangle is isosceles} \;\cdot\; \text{Linear pairs are supplementary} \;\cdot\; \text{Reflexive property} \;\cdot\; \text{Given} \;\cdot\; \text{Base angles of an isosceles triangle are congruent} \;\cdot\; \text{SAS} \;\cdot\; \text{Definition of an angle bisector}$ $\egin{aligned}&\textbf{Given: } T$ is the midpoint of $\overline{P S}$ and $\overline{QR} \\&\textbf{Prove: } \overline{PQ} \cong \overline{SR}\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{\|l\|l\|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; T \text{ is the midpoint of } \overline{PS} \text{ and } \overline{QR} & \text{Given} \\\hline 2.\; \overline{PT} \cong \overline{ST} & \; \\\hline 3.\; \overline{QT} \cong \overline{RT} & \text{Definition of midpoint} \\\hline 4.\; \angle PTQ \cong \angle STR & \; \\\hline 5.\; \triangle PQT \cong \triangle SRT & \; \\\hline 6.\; \overline{PQ} \cong \overline{SR} & \text{CPCTC} \\\hline\end{array}$ $\egin{aligned}&\textbf{Given: } T$ is the midpoint of $\overline{P S}$ and $\overline{QR} \\&\textbf{Prove: } \overline{PQ} \cong \overline{SR}\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{\|l\|l\|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; T \text{ is the midpoint of } \overline{PS} \text{ and } \overline{QR} & \; \\\hline 2.\; \overline{PT} \cong \overline{ST} & \; \\\hline 3.\; \overline{QT} \cong \overline{RT} & \; \\\hline 4.\; \angle PTQ \cong \angle STR & \; \\\hline 5.\; \triangle PQT \cong \triangle SRT & \; \\\hline 6.\; \overline{PQ} \cong \overline{SR} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{CPCTC} \;\cdot\; \text{Vertical angles are congruent} \;\cdot\; \text{Given} \;\cdot\; \text{AA similarity} \;\cdot\; \text{Definition of midpoint} \;\cdot\; \text{Angle addition postulate} \;\cdot\; \text{AAS} \;\cdot\; \text{SAS} \;\cdot\; \text{Alternate interior angles of parallel lines are congruent}$ $\egin{aligned}&\textbf{Given: } T$ is the midpoint of $\overline{P S}$ and $\overline{QR} \\&\textbf{Prove: } \overline{PQ} \cong \overline{SR}\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{\|l\|l\|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; T \text{ is the midpoint of } \overline{PS} \text{ and } \overline{QR} & \text{Given} \\\hline 2.\; \overline{PT} \cong \overline{ST} & \; \\\hline 3.\; \overline{QT} \cong \overline{RT} & \text{Definition of midpoint} \\\hline 4.\; \angle PTQ \cong \angle STR & \text{Vertical angles are congruent} \\\hline 5.\; \triangle PQT \cong \triangle SRT & \; \\\hline 6.\; \overline{PQ} \cong \overline{SR} & \; \\\hline\end{array}$ $\egin{aligned}&\textbf{Given: } P$ is the midpoint of $\overline{K N}$ and $\overline{LM} \\&\textbf{Prove: } \overline{KL} \cong \overline{NM}\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{\|l\|l\|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; P \text{ is the midpoint of } \overline{KN} \text{ and } \overline{LM} & \; \\\hline 2.\; \overline{KP} \cong \overline{NP} & \; \\\hline 3.\; \overline{LP} \cong \overline{MP} & \; \\\hline 4.\; \angle KPL \cong \angle NPM & \; \\\hline 5.\; \triangle KLP \cong \triangle NMP & \; \\\hline 6.\; \overline{KL} \cong \overline{NM} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Definition of midpoint} \;\cdot\; \text{Vertical angles are congruent} \;\cdot\; \text{Given} \;\cdot\; \text{SAS} \;\cdot\; \text{All right angles are congruent} \;\cdot\; \text{All radii of a circle are congruent} \;\cdot\; \text{CPCTC} \;\cdot\; \text{Definition of a median} \;\cdot\; \text{HL}$ $\egin{aligned}&\textbf{Given: } \overline{JK} \cong \overline{LK} \\ &\qquad\quad\; \overline{KM}$ bisects $\angle JKL \\&\textbf{Prove: } \angle J \cong \angle L\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{\|l\|l\|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overline{JK} \cong \overline{LK} & \text{Given} \\\hline 2.\; \overline{KM} \text{ bisects } \angle JKL & \text{Given} \\\hline 3.\; \angle JKM \cong \angle LKM & \text{Definition of an angle bisector} \\\hline 4.\; \overline{KM} \cong \overline{KM} & \; \\\hline 5.\; \triangle JKM \cong \triangle LKM & \; \\\hline 6.\; \angle J \cong \angle L & \; \\\hline\end{array}$ $\egin{aligned}&\textbf{Given: } J$ is the midpoint of $\overline{F I}$ and $\overline{GH} \\&\textbf{Prove: } \overline{FG} \cong \overline{IH}\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{\|l\|l\|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; J \text{ is the midpoint of } \overline{FI} \text{ and } \overline{GH} & \; \\\hline 2.\; \overline{FJ} \cong \overline{IJ} & \; \\\hline 3.\; \overline{GJ} \cong \overline{HJ} & \; \\\hline 4.\; \angle FJG \cong \angle IJH & \; \\\hline 5.\; \triangle FGJ \cong \triangle IHJ & \; \\\hline 6.\; \overline{FG} \cong \overline{IH} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{CPCTC} \;\cdot\; \text{If base angles are congruent, the triangle is isosceles} \;\cdot\; \text{If both pairs of opposite sides are parallel, the figure is a parallelogram} \;\cdot\; \text{Subtraction property} \;\cdot\; \text{Angle addition postulate} \;\cdot\; \text{Definition of midpoint} \;\cdot\; \text{Vertical angles are congruent} \;\cdot\; \text{Given} \;\cdot\; \text{SAS}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

CPCTC Proofs

Date Period

Complete each proof.

$\egin{aligned}&\textbf{Given: } \overline{AB} \cong \overline{CB} \\ &\qquad\quad\; \overline{BD}$ bisects $\angle ABC \\&\textbf{Prove: } \angle A \cong \angle C\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overline{AB} \cong \overline{CB} & \text{Given} \\\hline 2.\; \overline{BD} \text{ bisects } \angle ABC & \text{Given} \\\hline 3.\; \angle ABD \cong \angle CBD & \; \\\hline 4.\; \overline{BD} \cong \overline{BD} & \; \\\hline 5.\; \triangle ABD \cong \triangle CBD & \text{SAS} \\\hline 6.\; \angle A \cong \angle C & \; \\\hline\end{array}$
$\egin{aligned}&\textbf{Given: } \overline{RS} \cong \overline{TS} \\ &\qquad\quad\; \overline{SU}$ bisects $\angle RST \\&\textbf{Prove: } \angle R \cong \angle T\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overline{RS} \cong \overline{TS} & \; \\\hline 2.\; \overline{SU} \text{ bisects } \angle RST & \; \\\hline 3.\; \angle RSU \cong \angle TSU & \; \\\hline 4.\; \overline{SU} \cong \overline{SU} & \; \\\hline 5.\; \triangle RSU \cong \triangle TSU & \; \\\hline 6.\; \angle R \cong \angle T & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Supplements of congruent angles are congruent} \;\cdot\; \text{CPCTC} \;\cdot\; \text{If base angles are congruent, the triangle is isosceles} \;\cdot\; \text{Linear pairs are supplementary} \;\cdot\; \text{Reflexive property} \;\cdot\; \text{Given} \;\cdot\; \text{Base angles of an isosceles triangle are congruent} \;\cdot\; \text{SAS} \;\cdot\; \text{Definition of an angle bisector}$
$\egin{aligned}&\textbf{Given: } T$ is the midpoint of $\overline{P S}$ and $\overline{QR} \\&\textbf{Prove: } \overline{PQ} \cong \overline{SR}\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; T \text{ is the midpoint of } \overline{PS} \text{ and } \overline{QR} & \text{Given} \\\hline 2.\; \overline{PT} \cong \overline{ST} & \; \\\hline 3.\; \overline{QT} \cong \overline{RT} & \text{Definition of midpoint} \\\hline 4.\; \angle PTQ \cong \angle STR & \; \\\hline 5.\; \triangle PQT \cong \triangle SRT & \; \\\hline 6.\; \overline{PQ} \cong \overline{SR} & \text{CPCTC} \\\hline\end{array}$
$\egin{aligned}&\textbf{Given: } T$ is the midpoint of $\overline{P S}$ and $\overline{QR} \\&\textbf{Prove: } \overline{PQ} \cong \overline{SR}\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; T \text{ is the midpoint of } \overline{PS} \text{ and } \overline{QR} & \; \\\hline 2.\; \overline{PT} \cong \overline{ST} & \; \\\hline 3.\; \overline{QT} \cong \overline{RT} & \; \\\hline 4.\; \angle PTQ \cong \angle STR & \; \\\hline 5.\; \triangle PQT \cong \triangle SRT & \; \\\hline 6.\; \overline{PQ} \cong \overline{SR} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{CPCTC} \;\cdot\; \text{Vertical angles are congruent} \;\cdot\; \text{Given} \;\cdot\; \text{AA similarity} \;\cdot\; \text{Definition of midpoint} \;\cdot\; \text{Angle addition postulate} \;\cdot\; \text{AAS} \;\cdot\; \text{SAS} \;\cdot\; \text{Alternate interior angles of parallel lines are congruent}$
$\egin{aligned}&\textbf{Given: } T$ is the midpoint of $\overline{P S}$ and $\overline{QR} \\&\textbf{Prove: } \overline{PQ} \cong \overline{SR}\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; T \text{ is the midpoint of } \overline{PS} \text{ and } \overline{QR} & \text{Given} \\\hline 2.\; \overline{PT} \cong \overline{ST} & \; \\\hline 3.\; \overline{QT} \cong \overline{RT} & \text{Definition of midpoint} \\\hline 4.\; \angle PTQ \cong \angle STR & \text{Vertical angles are congruent} \\\hline 5.\; \triangle PQT \cong \triangle SRT & \; \\\hline 6.\; \overline{PQ} \cong \overline{SR} & \; \\\hline\end{array}$
$\egin{aligned}&\textbf{Given: } P$ is the midpoint of $\overline{K N}$ and $\overline{LM} \\&\textbf{Prove: } \overline{KL} \cong \overline{NM}\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; P \text{ is the midpoint of } \overline{KN} \text{ and } \overline{LM} & \; \\\hline 2.\; \overline{KP} \cong \overline{NP} & \; \\\hline 3.\; \overline{LP} \cong \overline{MP} & \; \\\hline 4.\; \angle KPL \cong \angle NPM & \; \\\hline 5.\; \triangle KLP \cong \triangle NMP & \; \\\hline 6.\; \overline{KL} \cong \overline{NM} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Definition of midpoint} \;\cdot\; \text{Vertical angles are congruent} \;\cdot\; \text{Given} \;\cdot\; \text{SAS} \;\cdot\; \text{All right angles are congruent} \;\cdot\; \text{All radii of a circle are congruent} \;\cdot\; \text{CPCTC} \;\cdot\; \text{Definition of a median} \;\cdot\; \text{HL}$
$\egin{aligned}&\textbf{Given: } \overline{JK} \cong \overline{LK} \\ &\qquad\quad\; \overline{KM}$ bisects $\angle JKL \\&\textbf{Prove: } \angle J \cong \angle L\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overline{JK} \cong \overline{LK} & \text{Given} \\\hline 2.\; \overline{KM} \text{ bisects } \angle JKL & \text{Given} \\\hline 3.\; \angle JKM \cong \angle LKM & \text{Definition of an angle bisector} \\\hline 4.\; \overline{KM} \cong \overline{KM} & \; \\\hline 5.\; \triangle JKM \cong \triangle LKM & \; \\\hline 6.\; \angle J \cong \angle L & \; \\\hline\end{array}$
$\egin{aligned}&\textbf{Given: } J$ is the midpoint of $\overline{F I}$ and $\overline{GH} \\&\textbf{Prove: } \overline{FG} \cong \overline{IH}\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; J \text{ is the midpoint of } \overline{FI} \text{ and } \overline{GH} & \; \\\hline 2.\; \overline{FJ} \cong \overline{IJ} & \; \\\hline 3.\; \overline{GJ} \cong \overline{HJ} & \; \\\hline 4.\; \angle FJG \cong \angle IJH & \; \\\hline 5.\; \triangle FGJ \cong \triangle IHJ & \; \\\hline 6.\; \overline{FG} \cong \overline{IH} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{CPCTC} \;\cdot\; \text{If base angles are congruent, the triangle is isosceles} \;\cdot\; \text{If both pairs of opposite sides are parallel, the figure is a parallelogram} \;\cdot\; \text{Subtraction property} \;\cdot\; \text{Angle addition postulate} \;\cdot\; \text{Definition of midpoint} \;\cdot\; \text{Vertical angles are congruent} \;\cdot\; \text{Given} \;\cdot\; \text{SAS}$



Sigma Prep · sigmaprep.io/worksheets Name CPCTC ProofsAnswer keyVersion 1 Date Period Complete each proof. $\egin{aligned}&\textbf{Given: } \overline{AB} \cong \overline{CB} \\ &\qquad\quad\; \overline{BD}$ bisects $\angle ABC \\&\textbf{Prove: } \angle A \cong \angle C\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{\|l\|l\|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overline{AB} \cong \overline{CB} & \text{Given} \\\hline 2.\; \overline{BD} \text{ bisects } \angle ABC & \text{Given} \\\hline 3.\; \angle ABD \cong \angle CBD & \; \\\hline 4.\; \overline{BD} \cong \overline{BD} & \; \\\hline 5.\; \triangle ABD \cong \triangle CBD & \text{SAS} \\\hline 6.\; \angle A \cong \angle C & \; \\\hline\end{array}$ $\begin{aligned} 3. Definition of an angle bisector \\ 4. Reflexive property \\ 6. CPCTC \end{aligned}$ $\egin{aligned}&\textbf{Given: } \overline{RS} \cong \overline{TS} \\ &\qquad\quad\; \overline{SU}$ bisects $\angle RST \\&\textbf{Prove: } \angle R \cong \angle T\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{\|l\|l\|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overline{RS} \cong \overline{TS} & \; \\\hline 2.\; \overline{SU} \text{ bisects } \angle RST & \; \\\hline 3.\; \angle RSU \cong \angle TSU & \; \\\hline 4.\; \overline{SU} \cong \overline{SU} & \; \\\hline 5.\; \triangle RSU \cong \triangle TSU & \; \\\hline 6.\; \angle R \cong \angle T & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Supplements of congruent angles are congruent} \;\cdot\; \text{CPCTC} \;\cdot\; \text{If base angles are congruent, the triangle is isosceles} \;\cdot\; \text{Linear pairs are supplementary} \;\cdot\; \text{Reflexive property} \;\cdot\; \text{Given} \;\cdot\; \text{Base angles of an isosceles triangle are congruent} \;\cdot\; \text{SAS} \;\cdot\; \text{Definition of an angle bisector}$ $\begin{aligned} 1. Given \\ 2. Given \\ 3. Definition of an angle bisector \\ 4. Reflexive property \\ 5. SAS \\ 6. CPCTC \end{aligned}$ $\egin{aligned}&\textbf{Given: } T$ is the midpoint of $\overline{P S}$ and $\overline{QR} \\&\textbf{Prove: } \overline{PQ} \cong \overline{SR}\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{\|l\|l\|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; T \text{ is the midpoint of } \overline{PS} \text{ and } \overline{QR} & \text{Given} \\\hline 2.\; \overline{PT} \cong \overline{ST} & \; \\\hline 3.\; \overline{QT} \cong \overline{RT} & \text{Definition of midpoint} \\\hline 4.\; \angle PTQ \cong \angle STR & \; \\\hline 5.\; \triangle PQT \cong \triangle SRT & \; \\\hline 6.\; \overline{PQ} \cong \overline{SR} & \text{CPCTC} \\\hline\end{array}$ $\begin{aligned} 2. Definition of midpoint \\ 4. Vertical angles are congruent \\ 5. SAS \end{aligned}$ $\egin{aligned}&\textbf{Given: } T$ is the midpoint of $\overline{P S}$ and $\overline{QR} \\&\textbf{Prove: } \overline{PQ} \cong \overline{SR}\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{\|l\|l\|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; T \text{ is the midpoint of } \overline{PS} \text{ and } \overline{QR} & \; \\\hline 2.\; \overline{PT} \cong \overline{ST} & \; \\\hline 3.\; \overline{QT} \cong \overline{RT} & \; \\\hline 4.\; \angle PTQ \cong \angle STR & \; \\\hline 5.\; \triangle PQT \cong \triangle SRT & \; \\\hline 6.\; \overline{PQ} \cong \overline{SR} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{CPCTC} \;\cdot\; \text{Vertical angles are congruent} \;\cdot\; \text{Given} \;\cdot\; \text{AA similarity} \;\cdot\; \text{Definition of midpoint} \;\cdot\; \text{Angle addition postulate} \;\cdot\; \text{AAS} \;\cdot\; \text{SAS} \;\cdot\; \text{Alternate interior angles of parallel lines are congruent}$ $\begin{aligned} 1. Given \\ 2. Definition of midpoint \\ 3. Definition of midpoint \\ 4. Vertical angles are congruent \\ 5. SAS \\ 6. CPCTC \end{aligned}$ $\egin{aligned}&\textbf{Given: } T$ is the midpoint of $\overline{P S}$ and $\overline{QR} \\&\textbf{Prove: } \overline{PQ} \cong \overline{SR}\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{\|l\|l\|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; T \text{ is the midpoint of } \overline{PS} \text{ and } \overline{QR} & \text{Given} \\\hline 2.\; \overline{PT} \cong \overline{ST} & \; \\\hline 3.\; \overline{QT} \cong \overline{RT} & \text{Definition of midpoint} \\\hline 4.\; \angle PTQ \cong \angle STR & \text{Vertical angles are congruent} \\\hline 5.\; \triangle PQT \cong \triangle SRT & \; \\\hline 6.\; \overline{PQ} \cong \overline{SR} & \; \\\hline\end{array}$ $\begin{aligned} 2. Definition of midpoint \\ 5. SAS \\ 6. CPCTC \end{aligned}$ $\egin{aligned}&\textbf{Given: } P$ is the midpoint of $\overline{K N}$ and $\overline{LM} \\&\textbf{Prove: } \overline{KL} \cong \overline{NM}\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{\|l\|l\|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; P \text{ is the midpoint of } \overline{KN} \text{ and } \overline{LM} & \; \\\hline 2.\; \overline{KP} \cong \overline{NP} & \; \\\hline 3.\; \overline{LP} \cong \overline{MP} & \; \\\hline 4.\; \angle KPL \cong \angle NPM & \; \\\hline 5.\; \triangle KLP \cong \triangle NMP & \; \\\hline 6.\; \overline{KL} \cong \overline{NM} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Definition of midpoint} \;\cdot\; \text{Vertical angles are congruent} \;\cdot\; \text{Given} \;\cdot\; \text{SAS} \;\cdot\; \text{All right angles are congruent} \;\cdot\; \text{All radii of a circle are congruent} \;\cdot\; \text{CPCTC} \;\cdot\; \text{Definition of a median} \;\cdot\; \text{HL}$ $\begin{aligned} 1. Given \\ 2. Definition of midpoint \\ 3. Definition of midpoint \\ 4. Vertical angles are congruent \\ 5. SAS \\ 6. CPCTC \end{aligned}$ $\egin{aligned}&\textbf{Given: } \overline{JK} \cong \overline{LK} \\ &\qquad\quad\; \overline{KM}$ bisects $\angle JKL \\&\textbf{Prove: } \angle J \cong \angle L\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{\|l\|l\|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overline{JK} \cong \overline{LK} & \text{Given} \\\hline 2.\; \overline{KM} \text{ bisects } \angle JKL & \text{Given} \\\hline 3.\; \angle JKM \cong \angle LKM & \text{Definition of an angle bisector} \\\hline 4.\; \overline{KM} \cong \overline{KM} & \; \\\hline 5.\; \triangle JKM \cong \triangle LKM & \; \\\hline 6.\; \angle J \cong \angle L & \; \\\hline\end{array}$ $\begin{aligned} 4. Reflexive property \\ 5. SAS \\ 6. CPCTC \end{aligned}$ $\egin{aligned}&\textbf{Given: } J$ is the midpoint of $\overline{F I}$ and $\overline{GH} \\&\textbf{Prove: } \overline{FG} \cong \overline{IH}\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{\|l\|l\|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; J \text{ is the midpoint of } \overline{FI} \text{ and } \overline{GH} & \; \\\hline 2.\; \overline{FJ} \cong \overline{IJ} & \; \\\hline 3.\; \overline{GJ} \cong \overline{HJ} & \; \\\hline 4.\; \angle FJG \cong \angle IJH & \; \\\hline 5.\; \triangle FGJ \cong \triangle IHJ & \; \\\hline 6.\; \overline{FG} \cong \overline{IH} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{CPCTC} \;\cdot\; \text{If base angles are congruent, the triangle is isosceles} \;\cdot\; \text{If both pairs of opposite sides are parallel, the figure is a parallelogram} \;\cdot\; \text{Subtraction property} \;\cdot\; \text{Angle addition postulate} \;\cdot\; \text{Definition of midpoint} \;\cdot\; \text{Vertical angles are congruent} \;\cdot\; \text{Given} \;\cdot\; \text{SAS}$ $\begin{aligned} 1. Given \\ 2. Definition of midpoint \\ 3. Definition of midpoint \\ 4. Vertical angles are congruent \\ 5. SAS \\ 6. CPCTC \end{aligned}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

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Name

CPCTC ProofsAnswer keyVersion 1

Date Period

Complete each proof.

$\egin{aligned}&\textbf{Given: } \overline{AB} \cong \overline{CB} \\ &\qquad\quad\; \overline{BD}$ bisects $\angle ABC \\&\textbf{Prove: } \angle A \cong \angle C\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overline{AB} \cong \overline{CB} & \text{Given} \\\hline 2.\; \overline{BD} \text{ bisects } \angle ABC & \text{Given} \\\hline 3.\; \angle ABD \cong \angle CBD & \; \\\hline 4.\; \overline{BD} \cong \overline{BD} & \; \\\hline 5.\; \triangle ABD \cong \triangle CBD & \text{SAS} \\\hline 6.\; \angle A \cong \angle C & \; \\\hline\end{array}$ $\begin{aligned} 3. Definition of an angle bisector \\ 4. Reflexive property \\ 6. CPCTC \end{aligned}$
$\egin{aligned}&\textbf{Given: } \overline{RS} \cong \overline{TS} \\ &\qquad\quad\; \overline{SU}$ bisects $\angle RST \\&\textbf{Prove: } \angle R \cong \angle T\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overline{RS} \cong \overline{TS} & \; \\\hline 2.\; \overline{SU} \text{ bisects } \angle RST & \; \\\hline 3.\; \angle RSU \cong \angle TSU & \; \\\hline 4.\; \overline{SU} \cong \overline{SU} & \; \\\hline 5.\; \triangle RSU \cong \triangle TSU & \; \\\hline 6.\; \angle R \cong \angle T & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Supplements of congruent angles are congruent} \;\cdot\; \text{CPCTC} \;\cdot\; \text{If base angles are congruent, the triangle is isosceles} \;\cdot\; \text{Linear pairs are supplementary} \;\cdot\; \text{Reflexive property} \;\cdot\; \text{Given} \;\cdot\; \text{Base angles of an isosceles triangle are congruent} \;\cdot\; \text{SAS} \;\cdot\; \text{Definition of an angle bisector}$ $\begin{aligned} 1. Given \\ 2. Given \\ 3. Definition of an angle bisector \\ 4. Reflexive property \\ 5. SAS \\ 6. CPCTC \end{aligned}$
$\egin{aligned}&\textbf{Given: } T$ is the midpoint of $\overline{P S}$ and $\overline{QR} \\&\textbf{Prove: } \overline{PQ} \cong \overline{SR}\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; T \text{ is the midpoint of } \overline{PS} \text{ and } \overline{QR} & \text{Given} \\\hline 2.\; \overline{PT} \cong \overline{ST} & \; \\\hline 3.\; \overline{QT} \cong \overline{RT} & \text{Definition of midpoint} \\\hline 4.\; \angle PTQ \cong \angle STR & \; \\\hline 5.\; \triangle PQT \cong \triangle SRT & \; \\\hline 6.\; \overline{PQ} \cong \overline{SR} & \text{CPCTC} \\\hline\end{array}$ $\begin{aligned} 2. Definition of midpoint \\ 4. Vertical angles are congruent \\ 5. SAS \end{aligned}$
$\egin{aligned}&\textbf{Given: } T$ is the midpoint of $\overline{P S}$ and $\overline{QR} \\&\textbf{Prove: } \overline{PQ} \cong \overline{SR}\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; T \text{ is the midpoint of } \overline{PS} \text{ and } \overline{QR} & \; \\\hline 2.\; \overline{PT} \cong \overline{ST} & \; \\\hline 3.\; \overline{QT} \cong \overline{RT} & \; \\\hline 4.\; \angle PTQ \cong \angle STR & \; \\\hline 5.\; \triangle PQT \cong \triangle SRT & \; \\\hline 6.\; \overline{PQ} \cong \overline{SR} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{CPCTC} \;\cdot\; \text{Vertical angles are congruent} \;\cdot\; \text{Given} \;\cdot\; \text{AA similarity} \;\cdot\; \text{Definition of midpoint} \;\cdot\; \text{Angle addition postulate} \;\cdot\; \text{AAS} \;\cdot\; \text{SAS} \;\cdot\; \text{Alternate interior angles of parallel lines are congruent}$ $\begin{aligned} 1. Given \\ 2. Definition of midpoint \\ 3. Definition of midpoint \\ 4. Vertical angles are congruent \\ 5. SAS \\ 6. CPCTC \end{aligned}$
$\egin{aligned}&\textbf{Given: } T$ is the midpoint of $\overline{P S}$ and $\overline{QR} \\&\textbf{Prove: } \overline{PQ} \cong \overline{SR}\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; T \text{ is the midpoint of } \overline{PS} \text{ and } \overline{QR} & \text{Given} \\\hline 2.\; \overline{PT} \cong \overline{ST} & \; \\\hline 3.\; \overline{QT} \cong \overline{RT} & \text{Definition of midpoint} \\\hline 4.\; \angle PTQ \cong \angle STR & \text{Vertical angles are congruent} \\\hline 5.\; \triangle PQT \cong \triangle SRT & \; \\\hline 6.\; \overline{PQ} \cong \overline{SR} & \; \\\hline\end{array}$ $\begin{aligned} 2. Definition of midpoint \\ 5. SAS \\ 6. CPCTC \end{aligned}$
$\egin{aligned}&\textbf{Given: } P$ is the midpoint of $\overline{K N}$ and $\overline{LM} \\&\textbf{Prove: } \overline{KL} \cong \overline{NM}\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; P \text{ is the midpoint of } \overline{KN} \text{ and } \overline{LM} & \; \\\hline 2.\; \overline{KP} \cong \overline{NP} & \; \\\hline 3.\; \overline{LP} \cong \overline{MP} & \; \\\hline 4.\; \angle KPL \cong \angle NPM & \; \\\hline 5.\; \triangle KLP \cong \triangle NMP & \; \\\hline 6.\; \overline{KL} \cong \overline{NM} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Definition of midpoint} \;\cdot\; \text{Vertical angles are congruent} \;\cdot\; \text{Given} \;\cdot\; \text{SAS} \;\cdot\; \text{All right angles are congruent} \;\cdot\; \text{All radii of a circle are congruent} \;\cdot\; \text{CPCTC} \;\cdot\; \text{Definition of a median} \;\cdot\; \text{HL}$ $\begin{aligned} 1. Given \\ 2. Definition of midpoint \\ 3. Definition of midpoint \\ 4. Vertical angles are congruent \\ 5. SAS \\ 6. CPCTC \end{aligned}$
$\egin{aligned}&\textbf{Given: } \overline{JK} \cong \overline{LK} \\ &\qquad\quad\; \overline{KM}$ bisects $\angle JKL \\&\textbf{Prove: } \angle J \cong \angle L\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overline{JK} \cong \overline{LK} & \text{Given} \\\hline 2.\; \overline{KM} \text{ bisects } \angle JKL & \text{Given} \\\hline 3.\; \angle JKM \cong \angle LKM & \text{Definition of an angle bisector} \\\hline 4.\; \overline{KM} \cong \overline{KM} & \; \\\hline 5.\; \triangle JKM \cong \triangle LKM & \; \\\hline 6.\; \angle J \cong \angle L & \; \\\hline\end{array}$ $\begin{aligned} 4. Reflexive property \\ 5. SAS \\ 6. CPCTC \end{aligned}$
$\egin{aligned}&\textbf{Given: } J$ is the midpoint of $\overline{F I}$ and $\overline{GH} \\&\textbf{Prove: } \overline{FG} \cong \overline{IH}\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; J \text{ is the midpoint of } \overline{FI} \text{ and } \overline{GH} & \; \\\hline 2.\; \overline{FJ} \cong \overline{IJ} & \; \\\hline 3.\; \overline{GJ} \cong \overline{HJ} & \; \\\hline 4.\; \angle FJG \cong \angle IJH & \; \\\hline 5.\; \triangle FGJ \cong \triangle IHJ & \; \\\hline 6.\; \overline{FG} \cong \overline{IH} & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{CPCTC} \;\cdot\; \text{If base angles are congruent, the triangle is isosceles} \;\cdot\; \text{If both pairs of opposite sides are parallel, the figure is a parallelogram} \;\cdot\; \text{Subtraction property} \;\cdot\; \text{Angle addition postulate} \;\cdot\; \text{Definition of midpoint} \;\cdot\; \text{Vertical angles are congruent} \;\cdot\; \text{Given} \;\cdot\; \text{SAS}$ $\begin{aligned} 1. Given \\ 2. Definition of midpoint \\ 3. Definition of midpoint \\ 4. Vertical angles are congruent \\ 5. SAS \\ 6. CPCTC \end{aligned}$

How to do these

Given the triangles are congruent, why is $∠ B ≅ ∠ C$ ?

The triangles were proved congruent on the line above.
Angle B and angle C are matching parts of those triangles.

The answer is CPCTC.

Where students go wrong

Using CPCTC to prove the triangles congruent. It works the other way round: congruent triangles first, corresponding parts second.

Also called corresponding parts of congruent triangles, CPCTC worksheet, proofs using CPCTC or two column proofs CPCTC.

What you can put on this worksheet

Fill in the missing reasons: $\egin{aligned}&\textbf{Given: } \overline{WX} \cong \overline{YZ} \\ &\qquad\quad\; \overline{XY} \cong \overline{ZW} \\&\textbf{Prove: } \angle X \cong \angle Z\end{aligned} \\[4pt] \text{Give the missing reasons.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overline{WX} \cong \overline{YZ} & \text{Given} \\\hline 2.\; \overline{XY} \cong \overline{ZW} & \; \\\hline 3.\; \overline{WY} \cong \overline{WY} & \text{Reflexive property} \\\hline 4.\; \triangle WXY \cong \triangle YZW & \; \\\hline 5.\; \angle X \cong \angle Z & \text{CPCTC} \\\hline\end{array}$ → $\begin{aligned} 2. Given \\ 4. SSS \end{aligned}$
Fill in every reason: $\egin{aligned}&\textbf{Given: } \overline{WX} \cong \overline{YZ} \\ &\qquad\quad\; \overline{XY} \cong \overline{ZW} \\&\textbf{Prove: } \angle X \cong \angle Z\end{aligned} \\[4pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overline{WX} \cong \overline{YZ} & \; \\\hline 2.\; \overline{XY} \cong \overline{ZW} & \; \\\hline 3.\; \overline{WY} \cong \overline{WY} & \; \\\hline 4.\; \triangle WXY \cong \triangle YZW & \; \\\hline 5.\; \angle X \cong \angle Z & \; \\\hline\end{array} \\[3pt] \text{Word bank: } \small \text{Corresponding angles of parallel lines are congruent} \;\cdot\; \text{Given} \;\cdot\; \text{CPCTC} \;\cdot\; \text{Addition property} \;\cdot\; \text{Inscribed angles that intercept the same arc are congruent} \;\cdot\; \text{SSS} \;\cdot\; \text{Supplements of congruent angles are congruent} \;\cdot\; \text{Reflexive property}$ → $\begin{aligned} 1. Given \\ 2. Given \\ 3. Reflexive property \\ 4. SSS \\ 5. CPCTC \end{aligned}$
Fill in the missing statements: $\egin{aligned}&\textbf{Given: } \overline{WX} \cong \overline{YZ} \\ &\qquad\quad\; \overline{XY} \cong \overline{ZW} \\&\textbf{Prove: } \angle X \cong \angle Z\end{aligned} \\[4pt] \text{Give the missing statements.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overline{WX} \cong \overline{YZ} & \text{Given} \\\hline 2.\; \; & \text{Given} \\\hline 3.\; \overline{WY} \cong \overline{WY} & \text{Reflexive property} \\\hline 4.\; \; & \text{SSS} \\\hline 5.\; \angle X \cong \angle Z & \text{CPCTC} \\\hline\end{array}$ → $\begin{aligned} 2. \overline{X Y} ≅ \overline{Z W} \\ 4. △ W X Y ≅ △ Y Z W \end{aligned}$
Write the whole proof: $\egin{aligned}&\textbf{Given: } \overline{WX} \cong \overline{YZ} \\ &\qquad\quad\; \overline{XY} \cong \overline{ZW} \\&\textbf{Prove: } \angle X \cong \angle Z\end{aligned} \\[4pt] \text{Write the proof. It takes 5 lines.} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \; & \; \\\hline 2.\; \; & \; \\\hline 3.\; \; & \; \\\hline 4.\; \; & \; \\\hline 5.\; \; & \; \\\hline\end{array}$ → $\begin{aligned} 1. \overline{W X} ≅ \overline{Y Z} \\ Given \\ 2. \overline{X Y} ≅ \overline{Z W} \\ Given \\ 3. \overline{W Y} ≅ \overline{W Y} \\ Reflexive property \\ 4. △ W X Y ≅ △ Y Z W \\ SSS \\ 5. ∠ X ≅ ∠ Z \\ CPCTC \end{aligned}$
Name the postulate that finishes the proof: $\egin{aligned}&\textbf{Given: } \overline{WX} \cong \overline{YZ} \\ &\qquad\quad\; \overline{XY} \cong \overline{ZW} \\&\textbf{Prove: } \angle X \cong \angle Z\end{aligned} \\[4pt] \text{Which postulate proves the triangles congruent?} \\[2pt] \small\def\arraystretch{1.7}\egin{array}{|l|l|}\hline\ extbf{Statements} & \ extbf{Reasons} \\\hline1.\; \overline{WX} \cong \overline{YZ} & \text{Given} \\\hline 2.\; \overline{XY} \cong \overline{ZW} & \text{Given} \\\hline 3.\; \overline{WY} \cong \overline{WY} & \text{Reflexive property} \\\hline 4.\; \triangle WXY \cong \triangle YZW & \; \\\hline\end{array}$ → SSS

Questions about these worksheets

Yes. Every cpctc 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 cpctc 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 CPCTC to prove the triangles congruent. It works the other way round: congruent triangles first, corresponding parts second.

CPCTC proofs worksheet

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CPCTC Proofs

CPCTC ProofsAnswer keyVersion 1

How to do these

Where students go wrong

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