Postulates, Definitions, and Theorems (Chapter 4) Segment Addition Postulate (SAP) All segments AB and BC have unique real number measures AB and BC such that: ABCBC = AC if and only if B is between A and C on AC. Definition of Segment Congruence (Def.) AB y CD if and only if AB = CD. Definition of Segment Midpoint (Def.) B is the midpoint of AC if and only if B is between A and C on AC, and AB = BC. Segment Midpoint Theorem (SMT) If B is the midpoint of AC then AB y BC. Angle Measure Postulate (AMP) Every:ABC has a unique real number measure, m:abc, with 0 % m:abc % 180 + and such that: 1. A = C if and only if m:abc =0. 2.:ABC is formed by three distinct collinear points A, B, and C if and only if m:abc = 180 +. Angle Addition Postulate (AAP) m:abccm:cbd = m:abd if and only if C is an interior point of :ABD. Definition of Angle Congruence (Def.) :ABC y:def if and only if m:abc = m:def. Definition of Angle Bisector (Def.) BC bisects :ABD if and only if C is an interior point of:abd, and m:abc = m:cbd. Angle Bisector Theorem (ABT) BC bisects :ABD if and only if :ABC y:cbd. Congruence Substitution Theorem (Sub.) If two segments (two angles) are congruent then they may be substituted for each other in another formula. Definition of Special Angles and Segments (Def.) 1. :ABC is a right angle if and only if m:abc =90 +. 2. AB t if and only if :ABC is a right angle. 3. :ABC and :DEF are complementary if and only if m:abccm:def =90 +.
Definition of Special Angles and Segments - Continued (Def.) 4. :ABC and :DEF are supplementary if and only if m:abccm:def = 180 +. 5. :ABD and :DBC are a linear pair if and only if they are adjacent and supplementary. 6. :ABD and :EBC are vertical angles if and only if they are non-adjacent (opposite) angles formed by the two lines AC and DE that intersect in point B. Vertical Angles Theorem (VAT) If :ABD and :EBC are vertical angles then :ABC y:ebc. Angle Complements Theorem (ACT) Complements of the same angle are congruent. Angle Supplements Theorem (AST) Supplements of the same angle are congruent. Right Angle Theorem (RAT) Two intersecting perpendicular lines form four right angles. Perpendicular Line Postulate (PLP) Given a line and a point not on the line, there is exactly one line perpendicular to the given line and passing through the given point. Definition of Parallel Lines (Def.) Let l and m be distinct coplanar lines. l and m are parallel (l s m) if and only if they have no points in common, i.e. they don't intersect. Euclid's Parallel Postulate (EPP) Given a line and a point not on the line, there is exactly one line parallel to the given line and passing through the given point. Corresponding Angles Postulate (CAP) Corresponding angles are congruent if and only if l s m. Parallel Lines Theorem (PLT) 1. Alternate interior angles are congruent if and only if l s m. (AIA) 2. Alternate exterior angles are congruent if and only if l s m. (AEA) 3. Consecutive interior angles are supplementary if and only if l s m. (CIA) Perpendicular Transversal Theorem (PTT) 1. If l t t and m t t then l s m. 2. If l t t and l s m then m t t. Triangle Addition Theorem (TAT) For any triangle 6ABC, m:abc Cm:BCACm:CAB = 180 +.
Exterior Angle Theorem (EAT) For any triangle 6ABC, if D lies on the extension of side CA to ray CA, then m:dab = m:abccm:bca, i.e. the measure of the exterior angle is the sum of the measures of the two interior angles. Definition of Congruent Triangles (Def.) 6ABC y6def if and only if there is a 1-1 correspondence between the two triangles such that each pair of the corresponding pairs of sides are congruent and each pair of the corresponding pairs of angles are congruent. In proofs, we will often justify various steps with CPCTC which stands for Corresponding Parts of Congruent Triangles are Congruent. You may also just use: Def. of y 6's. Side-Angle-Side Postulate (SAS-P) Two triangles are congruent if and only if two sides and the included angle of one triangle are congruent to two corresponding sides and an included angle of another triangle. Congruent Triangles Theorem (CTT) 1. Two triangles are congruent if and only if all three sides are congruent. (SSS) 2. Two triangles are congruent if and only two angles and the included side of one triangle are congruent to two corresponding angles and an included side of another triangle. (ASA) 3. Two triangles are congruent if and only if two angles and a non-included side of one triangle are congruent to two corresponding angles and a non-included side of another triangle. (AAS) (Note that in some books these are all called postulates, but since these are actually all provable from SAS-P, we introduce them in a theorem. We will prove some of these in class.) Hypotenuse-Leg Theorem (HLT) Two right triangles are congruent if and only if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle. Definition of Special Triangles (Def.) 1. A triangle is isosceles if and only if at least two sides are congruent. 2. A triangle is equilateral if and only if all three sides are congruent. 3. A triangle is equiangular if and only if all three angles are congruent. Isosceles Triangle Theorem (ITT) Two sides of a triangle are congruent if and only if the angles opposite each of the two sides are congruent. Equilateral Triangle Theorem (ETT) The three sides of a triangle are congruent if and only if the three angles are congruent.