Geometry Unit 2 Collected Two-Column Proofs Study Guide

Unit Overview and Proof Fundamentals

Structure of Two-Column Proofs:
- Two-column proofs organize deductive geometric arguments into two corresponding columns:
- Statements (Left Column): A sequence of logically ordered mathematical assertions, equations, or geometric relationships leading from the premises to the final conclusion.
- Reasons (Right Column): The formal mathematical justification for each statement, consisting strictly of given information, definitions, postulates, algebraic properties of equality/congruence, or previously proven theorems.
- Diagrams are geometric representations and are not necessarily drawn to scale.
Fundamental Congruence Principle (CPCTC):
- Definition: CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent.
- Usage: Once two triangles are established as congruent via a triangle congruence postulate or theorem, any pair of corresponding sides or corresponding angles can be asserted as congruent by CPCTC.

Day 1: Introduction to Proof and Segment Congruence

Proof 1: Sum of Two Odd Integers
- Given: $p$ and $q$ are odd integers.
- Prove: $p + q$ is an even integer.
- Step-by-Step Proof:
- Statement 1: $p$ & $q$ are odd integers | Reason 1: Given.
- Statement 2: $p = 2m + 1$ and $m \in \mathbb{Z}$ | Reason 2: Definition of odd integer.
- Statement 3: $q = 2n + 1$ and $n \in \mathbb{Z}$ | Reason 3: Definition of odd integer.
- Statement 4: $p + q = p + q$ | Reason 4: Reflexive Property.
- Statement 5: $p + q = 2m + 1 + 2n + 1$ | Reason 5: Substitution Property.
- Statement 6: $p + q = 2(m + n + 1)$ | Reason 6: Simplify/factor.
- Statement 7: $m + n + 1 \in \mathbb{Z}$ | Reason 7: Closure of $\mathbb{Z}$ under addition.
- Statement 8: $p + q$ is an even integer | Reason 8: Definition of an even integer.
Proof 2: Definition of Midpoint
- Given: $M$ is the midpoint of $\overline{AB}$ .
- Prove: $AM = MB$ .
- Step-by-Step Proof:
- Statement 1: $M$ is the midpoint of $\overline{AB}$ | Reason 1: Given.
- Statement 2: $\overline{AM} \cong \overline{MB}$ | Reason 2: Definition of midpoint.
- Statement 3: $AM = MB$ | Reason 3: Definition of congruent segments.
Proof 3: Segment Addition and Congruence
- Given: Points $A$ , $B$ , $C$ , and $D$ lie on a line in that order; $\overline{AB} \cong \overline{CD}$ .
- Prove: $\overline{AC} \cong \overline{BD}$ .
- Step-by-Step Proof:
- Statement 1: $A$ , $B$ , $C$ , and $D$ lie on a line in that order | Reason 1: Given.
- Statement 2: $\overline{AB} \cong \overline{CD}$ | Reason 2: Given.
- Statement 3: $AB = CD$ | Reason 3: Definition of congruent segments.
- Statement 4: $AB + BC = AC$ | Reason 4: Segment Addition Postulate.
- Statement 5: $BC + CD = BD$ | Reason 5: Segment Addition Postulate.
- Statement 6: $CD + BC = AC$ | Reason 6: Substitution Property of Equality.
- Statement 7: $BC + CD = AC$ | Reason 7: Commutative Property of Addition.
- Statement 8: $AC = BD$ | Reason 8: Transitive Property of Equality (or Substitution Property).
- Statement 9: $\overline{AC} \cong \overline{BD}$ | Reason 9: Definition of congruent segments.

Day 2: Angle Addition, Subtraction, and Integer Products

Proof 4: Angle Addition Postulate
- Given: $\angle AOB \cong \angle COD$ ; rays $\overrightarrow{OB}$ and $\overrightarrow{OC}$ lie inside $\angle AOD$ in order.
- Prove: $\angle AOC \cong \angle BOD$ .
- Step-by-Step Proof:
- Statement 1: $\angle AOB \cong \angle COD$ | Reason 1: Given.
- Statement 2: $m\angle AOB = m\angle COD$ | Reason 2: Definition of congruent angles.
- Statement 3: $m\angle AOC = m\angle AOB + m\angle BOC$ | Reason 3: Angle Addition Postulate.
- Statement 4: $m\angle BOD = m\angle BOC + m\angle COD$ | Reason 4: Angle Addition Postulate.
- Statement 5: $m\angle AOC = m\angle COD + m\angle BOC$ | Reason 5: Substitution Property of Equality.
- Statement 6: $m\angle AOC = m\angle BOC + m\angle COD$ | Reason 6: Commutative Property of Addition.
- Statement 7: $m\angle AOC = m\angle BOD$ | Reason 7: Transitive Property of Equality (or Substitution Property).
- Statement 8: $\angle AOC \cong \angle BOD$ | Reason 8: Definition of congruent angles.
Proof 5: Product of Two Odd Integers
- Given: $a$ and $b$ are odd integers.
- Prove: $ab$ is an odd integer.
- Step-by-Step Proof:
- Statement 1: $a$ and $b$ are odd integers | Reason 1: Given.
- Statement 2: $a = 2m + 1$ and $b = 2n + 1$ , $m, n \in \mathbb{Z}$ | Reason 2: Definition of odd integer.
- Statement 3: $ab = (2m + 1)(2n + 1)$ | Reason 3: Substitution Property.
- Statement 4: $ab = 4mn + 2m + 2n + 1$ | Reason 4: Distributive Property.
- Statement 5: $ab = 2(2mn + m + n) + 1$ | Reason 5: Distributive Property / Factoring.
- Statement 6: $2mn + m + n \in \mathbb{Z}$ | Reason 6: Closure of integers under multiplication and addition.
- Statement 7: $ab$ is an odd integer | Reason 7: Definition of an odd integer.
Proof 6: Product of an Odd and an Even Integer
- Given: $p$ is odd and $q$ is even.
- Prove: $pq$ is an even integer.
- Step-by-Step Proof:
- Statement 1: $p$ is odd and $q$ is even | Reason 1: Given.
- Statement 2: $p = 2m + 1$ and $q = 2n$ , $m, n \in \mathbb{Z}$ | Reason 2: Definition of odd and even integers.
- Statement 3: $pq = (2m + 1)(2n)$ | Reason 3: Substitution Property.
- Statement 4: $pq = 4mn + 2n$ | Reason 4: Distributive Property / Multiplication.
- Statement 5: $pq = 2(2mn + n)$ | Reason 5: Factoring / Distributive Property.
- Statement 6: $2mn + n \in \mathbb{Z}$ | Reason 6: Closure of integers.
- Statement 7: $pq$ is an even integer | Reason 7: Definition of an even integer.
Proof 7: Segment Addition: A Converse-Style Result
- Given: Points $A, B, C, D$ are collinear in that order; $\overline{AC} \cong \overline{BD}$ .
- Prove: $\overline{AB} \cong \overline{CD}$ .
- Step-by-Step Proof:
- Statement 1: $A, B, C, D$ are collinear in that order | Reason 1: Given.
- Statement 2: $\overline{AC} \cong \overline{BD}$ | Reason 2: Given.
- Statement 3: $AC = BD$ | Reason 3: Definition of congruent segments.
- Statement 4: $AC = AB + BC$ | Reason 4: Segment Addition Postulate.
- Statement 5: $BD = BC + CD$ | Reason 5: Segment Addition Postulate.
- Statement 6: $AB + BC = BC + CD$ | Reason 6: Substitution Property of Equality.
- Statement 7: $AB = CD$ | Reason 7: Subtraction Property of Equality.
- Statement 8: $\overline{AB} \cong \overline{CD}$ | Reason 8: Definition of congruent segments.
Proof 8: Angle Addition: Subtraction of a Shared Angle
- Given: $\angle AOC \cong \angle BOD$ and rays $\overrightarrow{OB}, \overrightarrow{OC}$ lie inside $\angle AOD$ in order.
- Prove: $\angle AOB \cong \angle COD$ .
- Step-by-Step Proof:
- Statement 1: $\angle AOC \cong \angle BOD$ | Reason 1: Given.
- Statement 2: $m\angle AOC = m\angle BOD$ | Reason 2: Definition of congruent angles.
- Statement 3: $m\angle AOC = m\angle AOB + m\angle BOC$ | Reason 3: Angle Addition Postulate.
- Statement 4: $m\angle BOD = m\angle BOC + m\angle COD$ | Reason 4: Angle Addition Postulate.
- Statement 5: $m\angle AOB + m\angle BOC = m\angle BOC + m\angle COD$ | Reason 5: Substitution Property.
- Statement 6: $m\angle AOB = m\angle COD$ | Reason 6: Subtraction Property of Equality.
- Statement 7: $\angle AOB \cong \angle COD$ | Reason 7: Definition of congruent angles.

Day 3: Vertical Angles, Linear Pairs, and Bisectors

Proof 9: Vertical Angles Theorem
- Given: $\overline{AC}$ and $\overline{BD}$ intersect at $O$ .
- Prove: $\angle AOB \cong \angle COD$ .
- Step-by-Step Proof:
- Statement 1: $A, O, C$ are collinear on $\overline{AC}$ | Reason 1: Given.
- Statement 2: $B, O, D$ are collinear on $\overline{BD}$ | Reason 2: Given.
- Statement 3: $\angle AOB$ and $\angle BOC$ form a linear pair | Reason 3: Definition of linear pair.
- Statement 4: $\angle BOC$ and $\angle COD$ form a linear pair | Reason 4: Definition of linear pair.
- Statement 5: $m\angle AOB + m\angle BOC = 180^\circ$ | Reason 5: Linear Pair Postulate (Definition of supplementary angles).
- Statement 6: $m\angle BOC + m\angle COD = 180^\circ$ | Reason 6: Linear Pair Postulate.
- Statement 7: $m\angle AOB + m\angle BOC = m\angle BOC + m\angle COD$ | Reason 7: Transitive Property of Equality (or Substitution Property).
- Statement 8: $m\angle AOB = m\angle COD$ | Reason 8: Subtraction Property of Equality.
- Statement 9: $\angle AOB \cong \angle COD$ | Reason 9: Definition of congruent angles.
Proof 10: Vertical Angles at Two Intersections
- Given: $\overline{AC}$ and $\overline{BD}$ intersect at $O$ ; $\overline{EG}$ and $\overline{FH}$ intersect at $P$ ; $\angle AOB \cong \angle EPF$ .
- Prove: $\angle COD \cong \angle GPH$ .
- Step-by-Step Proof:
- Statement 1: $\overline{AC}$ and $\overline{BD}$ intersect at $O$ | Reason 1: Given.
- Statement 2: $\overline{EG}$ and $\overline{FH}$ intersect at $P$ | Reason 2: Given.
- Statement 3: $\angle AOB \cong \angle EPF$ | Reason 3: Given.
- Statement 4: $\angle AOB \cong \angle COD$ | Reason 4: Vertical Angles Theorem.
- Statement 5: $\angle EPF \cong \angle GPH$ | Reason 5: Vertical Angles Theorem.
- Statement 6: $\angle COD \cong \angle EPF$ | Reason 6: Transitive Property of Congruence (or Substitution Property).
- Statement 7: $\angle COD \cong \angle GPH$ | Reason 7: Transitive Property of Congruence.
Proof 11: Two Consecutive Angle Bisectors
- Given: $\overrightarrow{VX}$ bisects $\angle WVY$ ; $\overrightarrow{VY}$ bisects $\angle XVZ$ .
- Prove: $\angle WVX \cong \angle YVZ$ .
- Step-by-Step Proof:
- Statement 1: $\overrightarrow{VX}$ bisects $\angle WVY$ | Reason 1: Given.
- Statement 2: $\overrightarrow{VY}$ bisects $\angle XVZ$ | Reason 2: Given.
- Statement 3: $\angle WVX \cong \angle XVY$ | Reason 3: Definition of angle bisector.
- Statement 4: $\angle XVY \cong \angle YVZ$ | Reason 4: Definition of angle bisector.
- Statement 5: $\angle WVX \cong \angle YVZ$ | Reason 5: Transitive Property of Congruence.
Proof 12: Angle Bisector with Vertical Angles
- Given: $\overline{AD}$ and $\overline{BE}$ intersect at $O$ ; $\overrightarrow{OB}$ bisects $\angle AOC$ .
- Prove: $\angle BOC \cong \angle DOE$ .
- Step-by-Step Proof:
- Statement 1: $\overline{AD}$ and $\overline{BE}$ intersect at $O$ | Reason 1: Given.
- Statement 2: $\overrightarrow{OB}$ bisects $\angle AOC$ | Reason 2: Given.
- Statement 3: $\angle BOC \cong \angle AOB$ | Reason 3: Definition of angle bisector.
- Statement 4: $\angle AOB \cong \angle DOE$ | Reason 4: Vertical Angles Theorem.
- Statement 5: $\angle BOC \cong \angle DOE$ | Reason 5: Transitive Property of Congruence.
Proof 13: Congruent Linear-Pair Angles Imply Perpendicularity
- Given: Points $A, O, B$ are collinear in that order; $\angle AOC \cong \angle COB$ .
- Prove: $\overrightarrow{OC} \perp \overleftrightarrow{AB}$ .
- Step-by-Step Proof:
- Statement 1: $A, O, B$ are collinear in that order | Reason 1: Given.
- Statement 2: $\angle AOC \cong \angle COB$ | Reason 2: Given.
- Statement 3: $\angle AOC$ and $\angle COB$ form a linear pair | Reason 3: Definition of linear pair.
- Statement 4: $m\angle AOC + m\angle COB = 180^\circ$ | Reason 4: Linear Pair Postulate.
- Statement 5: $m\angle AOC = m\angle COB$ | Reason 5: Definition of congruent angles.
- Statement 6: $2m\angle AOC = 180^\circ$ | Reason 6: Substitution Property of Equality.
- Statement 7: $m\angle AOC = 90^\circ$ | Reason 7: Division Property of Equality.
- Statement 8: $\overrightarrow{OC} \perp \overleftrightarrow{AB}$ | Reason 8: Definition of perpendicular lines / rays.
Proof 14: Bisectors of Vertical Angles
- Given: $\overline{AD}$ and $\overline{BE}$ intersect at $O$ ; $\overrightarrow{OC}$ bisects $\angle AOB$ ; $\overrightarrow{OF}$ bisects $\angle DOE$ .
- Prove: $\angle AOC \cong \angle DOF$ .
- Step-by-Step Proof:
- Statement 1: $\overline{AD}$ and $\overline{BE}$ intersect at $O$ | Reason 1: Given.
- Statement 2: $\overrightarrow{OC}$ bisects $\angle AOB$ | Reason 2: Given.
- Statement 3: $\overrightarrow{OF}$ bisects $\angle DOE$ | Reason 3: Given.
- Statement 4: $\angle AOB \cong \angle DOE$ | Reason 4: Vertical Angles Theorem.
- Statement 5: $m\angle AOB = m\angle DOE$ | Reason 5: Definition of congruent angles.
- Statement 6: $m\angle AOC = \frac{1}{2}m\angle AOB$ | Reason 6: Definition of angle bisector.
- Statement 7: $m\angle DOF = \frac{1}{2}m\angle DOE$ | Reason 7: Definition of angle bisector.
- Statement 8: $m\angle AOC = \frac{1}{2}m\angle DOE$ | Reason 8: Substitution Property of Equality.
- Statement 9: $m\angle AOC = m\angle DOF$ | Reason 9: Transitive Property of Equality (or Substitution Property).
- Statement 10: $\angle AOC \cong \angle DOF$ | Reason 10: Definition of congruent angles.

Day 4: Parallel Lines, Transversals, and Angle Theorems

Diagram Reference for Parallel Lines and Transversals:
- Line $p$ is parallel to line $q$ ( $p \parallel q$ ), intersected by transversal $t$ at points $P$ and $Q$ .
- Angles at vertex $P$ : top-left is $\angle 1$ , top-right is $\angle 2$ , bottom-left is $\angle 3$ , bottom-right is $\angle 4$ .
- Angles at vertex $Q$ : top-left is $\angle 5$ , top-right is $\angle 6$ , bottom-left is $\angle 7$ , bottom-right is $\angle 8$ .
Proof 15: Corresponding Angles Theorem via Translation
- Given: $p \parallel q$ ; transversal $t$ intersects $p$ at $P$ and $q$ at $Q$ .
- Prove: $\angle 2 \cong \angle 6$ .
- Step-by-Step Proof:
- Statement 1: $p \parallel q$ | Reason 1: Given.
- Statement 2: $t$ intersects $p$ at $P$ | Reason 2: Given.
- Statement 3: $t$ intersects $q$ at $Q$ | Reason 3: Given.
- Statement 4: Let $T_{\overrightarrow{PQ}}$ be the translation by $\overrightarrow{PQ}$ | Reason 4: Definition of the rigid transformation.
- Statement 5: $T_{\overrightarrow{PQ}}(\angle 2) = \angle 6$ | Reason 5: Properties of translations and parallel lines.
- Statement 6: $\angle 2 \cong \angle 6$ | Reason 6: Translations preserve angle measures.
Proof 16: Alternate Interior Angles Theorem
- Given: $p \parallel q$ ; $t$ intersects both lines.
- Prove: $\angle 3 \cong \angle 6$ .
- Step-by-Step Proof:
- Statement 1: $p \parallel q$ and $t$ intersects both $p$ and $q$ | Reason 1: Given.
- Statement 2: $\angle 3 \cong \angle 2$ | Reason 2: Vertical Angles Theorem.
- Statement 3: $\angle 2 \cong \angle 6$ | Reason 3: Corresponding Angles Theorem.
- Statement 4: $\angle 3 \cong \angle 6$ | Reason 4: Transitive Property of Congruence.
Proof 17: Same-Side Interior Angles Theorem
- Given: $p \parallel q$ ; $t$ intersects both lines.
- Prove: $\angle 3$ and $\angle 5$ are supplementary.
- Step-by-Step Proof:
- Statement 1: $p \parallel q$ and $t$ intersects both $p$ and $q$ | Reason 1: Given.
- Statement 2: $\angle 1$ and $\angle 3$ form a linear pair | Reason 2: Definition of linear pair.
- Statement 3: $m\angle 1 + m\angle 3 = 180^\circ$ | Reason 3: Linear Pair Postulate.
- Statement 4: $\angle 1 \cong \angle 5$ | Reason 4: Corresponding Angles Theorem.
- Statement 5: $m\angle 1 = m\angle 5$ | Reason 5: Definition of congruent angles.
- Statement 6: $m\angle 5 + m\angle 3 = 180^\circ$ | Reason 6: Substitution Property of Equality.
- Statement 7: $\angle 3$ and $\angle 5$ are supplementary | Reason 7: Definition of supplementary angles.
Proof 18: Corresponding Angles Converse
- Given: Transversal $t$ intersects lines $p, q$ and $\angle 2 \cong \angle 6$ .
- Prove: $p \parallel q$ .
- Step-by-Step Proof:
- Statement 1: $\angle 2 \cong \angle 6$ | Reason 1: Given.
- Statement 2: $\angle 3 \cong \angle 2$ | Reason 2: Vertical Angles Theorem.
- Statement 3: $\angle 3 \cong \angle 6$ | Reason 3: Transitive Property of Congruence.
- Statement 4: $p \parallel q$ | Reason 4: Alternate Interior Angles Converse.
Proof 19: Alternate Exterior Angles Theorem
- Given: $p \parallel q$ ; $t$ intersects both lines.
- Prove: $\angle 1 \cong \angle 8$ .
- Step-by-Step Proof:
- Statement 1: $p \parallel q$ ; $t$ intersects both lines | Reason 1: Given.
- Statement 2: $\angle 1 \cong \angle 5$ | Reason 2: Corresponding Angles Theorem.
- Statement 3: $\angle 5 \cong \angle 8$ | Reason 3: Vertical Angles Theorem.
- Statement 4: $\angle 1 \cong \angle 8$ | Reason 4: Transitive Property of Congruence.
Proof 20: Same-Side Interior Angles Converse
- Given: $\angle 3$ and $\angle 5$ are supplementary.
- Prove: $p \parallel q$ .
- Step-by-Step Proof:
- Statement 1: $\angle 3$ and $\angle 5$ are supplementary | Reason 1: Given.
- Statement 2: $m\angle 3 + m\angle 5 = 180^\circ$ | Reason 2: Definition of supplementary angles.
- Statement 3: $\angle 1$ and $\angle 3$ form a linear pair | Reason 3: Definition of linear pair.
- Statement 4: $m\angle 1 + m\angle 3 = 180^\circ$ | Reason 4: Linear Pair Postulate.
- Statement 5: $m\angle 1 + m\angle 3 = m\angle 3 + m\angle 5$ | Reason 5: Transitive Property of Equality.
- Statement 6: $m\angle 1 = m\angle 5$ | Reason 6: Subtraction Property of Equality.
- Statement 7: $\angle 1 \cong \angle 5$ | Reason 7: Definition of congruent angles.
- Statement 8: $p \parallel q$ | Reason 8: Corresponding Angles Converse.

Day 5: Side-Angle-Side (SAS) and CPCTC

Proof 21: Vertical Angles, SAS, and CPCTC
- Given: $\overline{AC}$ and $\overline{BD}$ intersect at $O$ ; $\overline{OA} \cong \overline{OC}$ ; $\overline{OB} \cong \overline{OD}$ .
- Prove: $\overline{AB} \cong \overline{CD}$ .
- Step-by-Step Proof:
- Statement 1: $\overline{AC}$ and $\overline{BD}$ intersect at $O$ | Reason 1: Given.
- Statement 2: $\overline{OA} \cong \overline{OC}$ | Reason 2: Given.
- Statement 3: $\overline{OB} \cong \overline{OD}$ | Reason 3: Given.
- Statement 4: $\angle AOB \cong \angle COD$ | Reason 4: Vertical Angles Theorem.
- Statement 5: $\triangle AOB \cong \triangle COD$ | Reason 5: SAS Congruence Postulate.
- Statement 6: $\overline{AB} \cong \overline{CD}$ | Reason 6: CPCTC.
Proof 22: Angle Bisector, SAS, and a Shared Side
- Given: $\overline{AB} \cong \overline{AC}$ ; $D \in \overline{BC}$ ; $\overrightarrow{AD}$ bisects $\angle BAC$ .
- Prove: $\overline{BD} \cong \overline{DC}$ .
- Step-by-Step Proof:
- Statement 1: $\overline{AB} \cong \overline{AC}$ and $\overrightarrow{AD}$ bisects $\angle BAC$ | Reason 1: Given.
- Statement 2: $\angle BAD \cong \angle DAC$ | Reason 2: Definition of angle bisector.
- Statement 3: $\overline{AD} \cong \overline{AD}$ | Reason 3: Reflexive Property of Congruence.
- Statement 4: $\triangle ABD \cong \triangle ACD$ | Reason 4: SAS Congruence Postulate.
- Statement 5: $\overline{BD} \cong \overline{DC}$ | Reason 5: CPCTC.

Day 6: Isosceles Triangles, Perpendicular Bisectors, and ASA

Proof 23: Base Angles Theorem in the Constructed Triangle
- Given: $\overline{AB} \cong \overline{AC}$ ; $\overleftrightarrow{AE}$ is a perpendicular bisector of $\overline{BC}$ at $M$ .
- Prove: $\angle ABC \cong \angle ACB$ .
- Step-by-Step Proof:
- Statement 1: $\overline{AB} \cong \overline{AC}$ | Reason 1: Given.
- Statement 2: $\overleftrightarrow{AE}$ is a perpendicular bisector of $\overline{BC}$ | Reason 2: Given.
- Statement 3: $\overleftrightarrow{AE}$ intersects $\overline{BC}$ at $M$ | Reason 3: Given.
- Statement 4: $\angle BAM \cong \angle CAM$ | Reason 4: Definition of perpendicular bisector / angle bisector in isosceles triangle.
- Statement 5: $\overline{AM} \cong \overline{AM}$ | Reason 5: Reflexive Property of Congruence.
- Statement 6: $\triangle BAM \cong \triangle CAM$ | Reason 6: SAS Congruence Postulate.
- Statement 7: $\angle B \cong \angle C$ (or $\angle ABC \cong \angle ACB$ ) | Reason 7: CPCTC.
Proof 24: Perpendicular Bisector and SAS
- Given: $M \in \overline{BC}$ ; $\overleftrightarrow{AM}$ is the perpendicular bisector of $\overline{BC}$ .
- Prove: $\overline{AB} \cong \overline{AC}$ .
- Step-by-Step Proof:
- Statement 1: $M$ lies on $\overline{BC}$ and $\overleftrightarrow{AM}$ is the perpendicular bisector of $\overline{BC}$ | Reason 1: Given.
- Statement 2: $M$ is the midpoint of $\overline{BC}$ , and $\overleftrightarrow{AM} \perp \overline{BC}$ | Reason 2: Definition of perpendicular bisector.
- Statement 3: $\overline{BM} \cong \overline{CM}$ | Reason 3: Definition of midpoint.
- Statement 4: $\angle BMA$ and $\angle CMA$ are right angles | Reason 4: Definition of perpendicular lines.
- Statement 5: $\angle BMA \cong \angle CMA$ | Reason 5: All right angles are congruent.
- Statement 6: $\overline{AM} \cong \overline{AM}$ | Reason 6: Reflexive Property of Congruence.
- Statement 7: $\triangle BMA \cong \triangle CMA$ | Reason 7: SAS Congruence Postulate.
- Statement 8: $\overline{AB} \cong \overline{AC}$ | Reason 8: CPCTC.

Day 7: Side-Side-Side (SSS) Congruence

Proof 25: Median to the Base of an Isosceles Triangle
- Given: $\overline{AB} \cong \overline{AC}$ ; $D$ is the midpoint of $\overline{BC}$ .
- Prove: $\overrightarrow{AD}$ bisects $\angle BAC$ .
- Step-by-Step Proof:
- Statement 1: $\overline{AB} \cong \overline{AC}$ | Reason 1: Given.
- Statement 2: $D$ is the midpoint of $\overline{BC}$ | Reason 2: Given.
- Statement 3: $\overline{BD} \cong \overline{DC}$ | Reason 3: Definition of midpoint.
- Statement 4: $\overline{AD} \cong \overline{AD}$ | Reason 4: Reflexive Property of Congruence.
- Statement 5: $\triangle ABD \cong \triangle ACD$ | Reason 5: SSS Congruence Postulate.
- Statement 6: $\angle BAD \cong \angle DAC$ | Reason 6: CPCTC.
- Statement 7: $\overrightarrow{AD}$ bisects $\angle BAC$ | Reason 7: Definition of angle bisector.

Day 8: Parallelogram Theorems and Their Converses

Proof 26: Opposite Sides of a Parallelogram
- Given: $ABCD$ is a parallelogram.
- Prove: $\overline{AB} \cong \overline{CD}$ and $\overline{BC} \cong \overline{DA}$ .
- Step-by-Step Proof:
- Statement 1: $ABCD$ is a parallelogram | Reason 1: Given.
- Statement 2: $\overline{AB} \parallel \overline{CD}$ and $\overline{BC} \parallel \overline{AD}$ | Reason 2: Definition of a parallelogram.
- Statement 3: Draw diagonal $\overline{AC}$ | Reason 3: Construction (Through any two points there exists exactly one line).
- Statement 4: $\angle BAC \cong \angle DCA$ | Reason 4: Alternate Interior Angles Theorem.
- Statement 5: $\angle BCA \cong \angle DAC$ | Reason 5: Alternate Interior Angles Theorem.
- Statement 6: $\overline{AC} \cong \overline{CA}$ | Reason 6: Reflexive Property of Congruence.
- Statement 7: $\triangle ABC \cong \triangle CDA$ | Reason 7: ASA Congruence Postulate.
- Statement 8: $\overline{AB} \cong \overline{CD}$ and $\overline{BC} \cong \overline{DA}$ | Reason 8: CPCTC.
Proof 27: Opposite Angles of a Parallelogram
- Given: $ABCD$ is a parallelogram.
- Prove: $\angle ABC \cong \angle CDA$ .
- Step-by-Step Proof:
- Statement 1: $ABCD$ is a parallelogram | Reason 1: Given.
- Statement 2: $\overline{AB} \cong \overline{CD}$ and $\overline{BC} \cong \overline{DA}$ | Reason 2: Opposite sides of a parallelogram are congruent (Proven in Proof 26).
- Statement 3: $\overline{AC} \cong \overline{CA}$ | Reason 3: Reflexive Property of Congruence.
- Statement 4: $\triangle ABC \cong \triangle CDA$ | Reason 4: SSS Congruence Postulate.
- Statement 5: $\angle ABC \cong \angle CDA$ | Reason 5: CPCTC.
Proof 28: Converse of the Opposite Sides Theorem
- Given: $\overline{AB} \cong \overline{CD}$ and $\overline{BC} \cong \overline{DA}$ .
- Prove: $ABCD$ is a parallelogram.
- Step-by-Step Proof:
- Statement 1: $\overline{AB} \cong \overline{CD}$ and $\overline{BC} \cong \overline{DA}$ | Reason 1: Given.
- Statement 2: Draw diagonal $\overline{AC}$ | Reason 2: Construction.
- Statement 3: $\overline{AC} \cong \overline{CA}$ | Reason 3: Reflexive Property of Congruence.
- Statement 4: $\triangle ABC \cong \triangle CDA$ | Reason 4: SSS Congruence Postulate.
- Statement 5: $\angle BAC \cong \angle DCA$ and $\angle BCA \cong \angle DAC$ | Reason 5: CPCTC.
- Statement 6: $\overline{AB} \parallel \overline{CD}$ and $\overline{BC} \parallel \overline{AD}$ | Reason 6: Alternate Interior Angles Converse.
- Statement 7: $ABCD$ is a parallelogram | Reason 7: Definition of a parallelogram.