Axiomatic Systems and Proofs

• A mathematical system is established through an axiomatic structure.
• An axiomatic structure consists of undefined terms, defined terms, postulates, and theorems.
• A proof is a logical statement where each statement is supported by a statement that is accepted as true.

Types of Proof

• Informal Proof/Paragraph: explains why a conjecture for a given situation is true in paragraph form.
• Formal Proof/Two-Column: has two columns, with statements in the first and the reasons for each statement in the second.
• Direct Proof: uses logical reasoning to directly reach a conclusion.
• Indirect Proof: assumes the original statement is false, proving it until finding a contradiction.

Geometric properties for writing proofs

• Betweenness: A-B-C if and only if AB + BC = AC
• Midpoint: A is the midpoint of segment BC if and only if AB = AC.
• Segment Bisector: Segment AB bisects segment PQ at B if and only if segment PB = segment QB.
• Right Angle: ∠A is a right angle if and only if m∠A = 90°.
• Acute Angle: ∠A is an acute angle if and only if m∠A < 90°.
• Obtuse Angle: ∠A is an obtuse angle if and only if m∠A > 90°.
• Perpendicular Line Segments: Segment AB ⊥ Segment AC if and only if ∠BAC is a right angle.
• Complementary Angles: ∠A and ∠B are complementary angles if and only if m∠A + m∠B = 90°.
• Supplementary Angles: ∠A and ∠B are supplementary angles if and only if m∠A + m∠B = 180°.
• Linear Pair: Includes two opposite rays PQ→ and PR→ and another ray PT→, where ∠QPT and ∠TPR form a linear pair.
• Angle Bisector: AD→ bisects ∠BAC if and only if ∠BAD = ∠DAC.
• Congruent Segments: Segment AB = Segment CD if and only if AB = CD.
• Congruent Angles: ∠A = ∠B if and only if m∠A = m∠B.

Properties of Equality

• Addition Property of Equality (APE): If a = b and c = d, then a + c = b + d, or if a = b, then a + c = b + c.
• Multiplication Property of Equality (MPE): If a = b and c = d, then ac = bd, or if a = b, then ac = bc.
• Subtraction Property of Equality (SPE): If a = b and c = d, then a - c = b - d, or if a = b, then a - c = b - c.
• Reflexive Property of Equality: For every real number a, a = a.
• Symmetric Property: If a = b, then b = a.
• Transitive Property of Equality: If a = b and b = c, then a = c.
• Law of Substitution: If a + b = c and b = x, then a + x = c.

Postulates

• The Angle Addition Postulate (AAP): If T is in the interior of ∠PQR, then m∠PQR = m∠PQT + m∠TQR.
• PCAC Postulate: If two parallel lines are cut by a transversal, then the corresponding angles are congruent.
• Supplement Postulate: If ∠1 and ∠2 form a linear pair, then ∠1 and ∠2 are supplementary angles.

Theorems

• Vertical Angle Theorem (VAT): If ∠1 and ∠2 are vertical angles, then ∠1 = ∠2.
• The Supplement Theorem (ST): Supplements of congruent angles are congruent.
• The Complement Theorem: Complements of congruent angles are congruent.
• PAIC Theorem: If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
• PSSIAS Theorem: If two parallel lines are cut by a transversal, then same-side interior angles are supplementary.
• The sum of the measures of the angle of a Triangle Theorem: In plane ABS, m∠A + m∠B + m∠C = 180°.
• The Exterior Angle Theorem (EAT): In triangle ABC, if ∠ACD is an exterior angle, then m∠ACD = m∠A + m∠B.

Postulates and Theorems Involving Angles Formed by Parallel Lines cut by a Transversal

• When two parallel lines are cut by a transversal, eight angles are formed, where any pair is either congruent or supplementary.
• Corresponding Angles are located on the same side of the transversal on different parallel lines and have equal measures.
• Alternate Interior Angles: located on the inside of two lines on opposite sides of the transversal and have equal measures.
• Alternate Exterior Angles: located on the outside of two lines on opposite sides of the transversal and have equal measures.
• Same-side Interior Angles: located on the inside of two lines on the same sides of the transversal and are supplementary..
• PAIC Theorem proof: If two parallel lines are cut by a transversal, alternate interior angles are congruent.
• PSSIAS Theorem proof: two parallel lines are cut by a transversal, then a pair of same side interior angles are supplementary.