Geometric Definitions, Postulates, and Algebraic Properties of Equality and Congruence

Foundations of Mathematical Definitions

  • Structure of Mathematical Definitions:

    • Mathematical definitions can be written as true biconditional statements.

    • A biconditional statement asserts an "if and only if" equivalence, meaning the conditional statement and its converse are both true simultaneously.

Geometric Definitions

  • Definition of (Segment) Congruence:

    • Segments are congruent if their lengths are equal.

    • Symbolic formulation: Segment AB‾\overline{AB} is congruent to segment CD‾\overline{CD} (AB‾≅CD‾\overline{AB} \cong \overline{CD}) if and only if their lengths are equal (AB=CDAB = CD).

  • Definition of (Angle) Congruence:

    • Angles are congruent if the measure of their angles are equal.

    • Symbolic formulation: ∠A≅∠B\angle A \cong \angle B if and only if m∠A=m∠Bm\angle A = m\angle B.

  • Definition of a Midpoint:

    • A point is the midpoint of a segment if it divides the segment into 22 congruent segments, or 22 segments with the same length.

    • If point MM is the midpoint of segment AB‾\overline{AB}, then AM‾≅MB‾\overline{AM} \cong \overline{MB} and AM=MBAM = MB.

  • Definition of a Segment Bisector:

    • A line, ray, or plane is a segment bisector if it intersects a segment at its midpoint.

    • The bisector divides the targeted segment into 22 congruent sub-segments at the point of intersection.

  • Definition of an Angle Bisector:

    • A ray or segment is an angle bisector if it divides an angle into 22 congruent angles, or 22 angles with the same measure.

    • If ray BD→\overrightarrow{BD} bisects ∠ABC\angle ABC, then ∠ABD≅∠DBC\angle ABD \cong \angle DBC and m∠ABD=m∠DBCm\angle ABD = m\angle DBC.

  • Definition of a Right Angle:

    • An angle is a right angle if the measure of the angle is 90∘90^\circ.

    • Symbolic formulation: ∠A\angle A is a right angle if and only if m∠A=90∘m\angle A = 90^\circ.

  • Definition of Complementary Angles:

    • Two angles are complementary if the measure of their angles add up to 90∘90^\circ.

    • Algebraic representation: Two angles ∠1\angle 1 and ∠2\angle 2 are complementary if and only if:

    • m∠1+m∠2=90∘m\angle 1 + m\angle 2 = 90^\circ

  • Definition of Supplementary Angles:

    • Two angles are supplementary if the measure of their angles add up to 180∘180^\circ.

    • Algebraic representation: Two angles ∠1\angle 1 and ∠2\angle 2 are supplementary if and only if:

    • m∠1+m∠2=180∘m\angle 1 + m\angle 2 = 180^\circ

  • Definition of Vertical Angles:

    • Two angles whose sides form 22 pairs of opposite rays.

    • Vertical angles are non-adjacent angles formed by the intersection of two straight lines.

  • Definition of a Linear Pair:

    • Two adjacent angles that form a straight line.

    • The non-common sides of a linear pair form a pair of opposite rays, extending in opposite directions along a single straight line.

Geometric Postulates


Geometric Postulates and Angle Addition Diagrams

  • Segment Addition Postulate:

    • Condition: If point BB is between point AA and point CC, then:

    • AB+BC=ACAB + BC = AC

    • Meaning: The total distance between the endpoints of a straight collinear segment equals the sum of the individual lengths of its non-overlapping segments.

  • Angle Addition Postulate:

    • Condition: If point BB lies in the interior of ∠AOC\angle AOC, then:

    • m∠AOB+m∠BOC=m∠AOCm\angle AOB + m\angle BOC = m\angle AOC

    • Meaning: The measure of a larger angle is equal to the sum of the measures of the two adjacent interior angles that compose it.

  • Linear Pair Postulate:

    • Statement: Linear pairs of angles are supplementary.

    • Consequence: If two adjacent angles form a linear pair along a straight line, the sum of their degree measures equals 180∘180^\circ.

Properties of Equality (POE)

  • Addition Property of Equality:

    • Condition and result: If a=ba = b and c=dc = d, then:

    • a+c=b+da + c = b + d

  • Subtraction Property of Equality:

    • Condition and result: If a=ba = b and c=dc = d, then:

    • a−c=b−da - c = b - d

  • Multiplication Property of Equality:

    • Condition and result: If a=ba = b, then:

    • ca=cbca = cb

  • Division Property of Equality:

    • Condition and result: If a=ba = b and c≠0c \neq 0, then:

    • ac=bc\frac{a}{c} = \frac{b}{c}

  • Reflexive Property of Equality:

    • Statement: Any quantity is equal to itself:

    • a=aa = a

  • Symmetric Property of Equality:

    • Condition and result: If a=ba = b, then:

    • b=ab = a

  • Transitive Property of Equality:

    • Condition and result: If a=ba = b and b=cb = c, then:

    • a=ca = c

  • Substitution Property of Equality:

    • Statement: A value or expression can be replaced by its equivalent in any equation.

    • Example: If a=x+ba = x + b and b=yb = y, then:

    • a=x+ya = x + y

Other Algebraic Properties

  • Distributive Property:

    • Statement: Multiplying a factor across a sum inside parentheses:

    • a(b+c)=ab+aca(b + c) = ab + ac

  • Additive Identity:

    • Statement: Adding zero to any value preserves its equality:

    • If a=ba = b, then:

    • a+0=ba + 0 = b

    • and

    • a=b+0a = b + 0

  • Multiplicative Identity:

    • Statement: Multiplying any value by 11 preserves its equality:

    • If a=ba = b, then:

    • a×1=ba \times 1 = b

    • and

    • a=b×1a = b \times 1

Properties of Congruence (POC)

  • Reflexive Property of Congruence:

    • Any geometric segment or angle is congruent to itself.

    • Line segments:

    • AB‾≅AB‾\overline{AB} \cong \overline{AB}

    • Angles:

    • ∠C≅∠C\angle C \cong \angle C

  • Symmetric Property of Congruence:

    • Congruence relations can be written in reverse order.

    • Line segments: If AB‾≅CD‾\overline{AB} \cong \overline{CD}, then:

    • CD‾≅AB‾\overline{CD} \cong \overline{AB}

    • Angles: If ∠C≅∠D\angle C \cong \angle D, then:

    • ∠D≅∠C\angle D \cong \angle C

  • Transitive Property of Congruence:

    • Geometric entities congruent to the same entity are congruent to each other.

    • Line segments: If AB‾≅CD‾\overline{AB} \cong \overline{CD} and CD‾≅EF‾\overline{CD} \cong \overline{EF}, then:

    • AB‾≅EF‾\overline{AB} \cong \overline{EF}

    • Angles: If ∠C≅∠D\angle C \cong \angle D and ∠D≅∠E\angle D \cong \angle E, then:

    • ∠C≅∠E\angle C \cong \angle E

  • Critical Restriction:

    • NOTE: There is NO Substitution Property of Congruence!!

    • Substitution can only be applied to equations of real numbers (measures and lengths) using the Substitution Property of Equality, not directly to geometric congruence statements.