Comprehensive Geometric and Algebraic Proofs Reference Guide

Properties of Equality

  • Usage Rule: Properties of equality can only be applied to mathematical statements that contain equal signs (==).

  • Addition Property of Equality:

    • Definition: Adding the same quantity to both sides of an equation yields an equivalent equation.

    • Formal Statement: If A=BA = B, then A+C=B+CA + C = B + C.

  • Subtraction Property of Equality:

    • Definition: Subtracting the same quantity from both sides of an equation yields an equivalent equation.

    • Formal Statement: If A=BA = B, then A−C=B−CA - C = B - C.

  • Multiplication Property of Equality:

    • Definition: Multiplying both sides of an equation by the same quantity yields an equivalent equation.

    • Formal Statement: If A=BA = B, then A×C=B×CA \times C = B \times C.

  • Division Property of Equality:

    • Definition: Dividing both sides of an equation by the same non-zero quantity yields an equivalent equation.

    • Formal Statement: If A=BA = B, then AC=BC\frac{A}{C} = \frac{B}{C}.

  • Distributive Property:

    • Definition: Multiplying a sum by a number gives the same result as multiplying each addend individually by the number and then adding the products together. It also applies in reverse for factoring common terms.

    • Formal Statement: a(b+c)=ab+aca(b + c) = ab + ac and ab+ac=x(a+b)ab + ac = x(a + b).

  • Substitution Property of Equality:

    • Definition: If two quantities are equal, then one may be replaced by the other in any expression or equation.

    • Formal Statement: If A=BA = B, then aa may be replaced by bb in any expression or equation.

  • Reflexive Property of Equality:

    • Definition: Any real value is always equal to itself.

    • Formal Statement: For any real number aa, A=AA = A.

  • Symmetric Property of Equality:

    • Definition: If a value equals a second value, the second value equals the first value.

    • Formal Statement: If A=BA = B, then B=AB = A.

  • Transitive Property of Equality:

    • Definition: If two quantities are each equal to a third quantity, then they are equal to each other.

    • Formal Statement: If A=BA = B and B=CB = C, then A=CA = C.

Properties of Congruence

  • Usage Rule: Properties of congruence can only be applied to mathematical statements that contain the congruence symbol (≅\cong).

  • Reflexive Property of Congruence:

    • Definition: A geometric figure is congruent to itself.

    • Segment Statement: For any segment ABAB, AB≅ABAB \cong AB.

  • Symmetric Property of Congruence:

    • Definition: If one geometric figure is congruent to a second figure, then the second figure is congruent to the first.

    • Segment Statement: If AB≅CDAB \cong CD, then CD≅ABCD \cong AB.

  • Transitive Property of Congruence:

    • Definition: If two geometric figures are congruent to a third figure, then they are congruent to each other.

    • Segment Statement: If AB≅CDAB \cong CD and CD≅EFCD \cong EF, then AB≅EFAB \cong EF.

Segment Proofs: Definitions and Postulates

  • Definition of Congruence:

    • Segments are congruent if and only if they have the exact same measure (equal lengths).

    • Equivalence Statement: If AB≅CDAB \cong CD, then AB=CDAB = CD. Conversely, if AB=CDAB = CD, then AB≅CDAB \cong CD.

  • Definition of Midpoint:

    • The midpoint of a segment divides the segment into two congruent parts with equal lengths.

    • Formal Statement: If MM is the midpoint of segment ABAB, then AM=MBAM = MB (or AM≅MBAM \cong MB).

  • Segment Addition Postulate:

    • Definition: If three points are collinear and one point lies between the other two, the sum of the lengths of the two smaller segments equals the length of the total segment.

    • Formal Statement: If AA, BB, and CC are collinear points and BB is between AA and CC, then AB+BC=ACAB + BC = AC.

Angle Proofs: Definitions, Postulates, and Theorems

  • Definition of Congruence:

    • The measures of two angles are equal if and only if the angles are congruent.

    • Formal Statement: m∠A=m∠B⇔∠A≅∠Bm\angle A = m\angle B \Leftrightarrow \angle A \cong \angle B.

  • Definition of a Right Angle:

    • An angle measures 90∘90^\circ if and only if it is a right angle.

    • Formal Statement: m∠A=90∘⇔∠A is a right anglem\angle A = 90^\circ \Leftrightarrow \angle A \text{ is a right angle}.

  • Definition of Complementary Angles:

    • Two angles are complementary if and only if the sum of their measures is 90∘90^\circ.

    • Relationship: Complementary →\rightarrow Sum is 90∘90^\circ.

  • Definition of Supplementary Angles:

    • Two angles are supplementary if and only if the sum of their measures is 180∘180^\circ.

    • Relationship: Supplementary →\rightarrow Sum is 180∘180^\circ.

  • Definition of an Angle Bisector:

    • An angle bisector is a ray or line that divides an angle into two equal parts.

  • Definition of Perpendicular:

    • Perpendicular lines intersect to form right angles.

  • Angle Addition Postulate:

    • Definition: If a ray lies in the interior of an angle, the measures of the two adjacent smaller angles add up to the measure of the entire angle.

    • Formal Statement: If point DD lies in the interior of ∠ABC\angle ABC, then m∠ABD+m∠DBC=m∠ABCm\angle ABD + m\angle DBC = m\angle ABC.

  • Vertical Angles Theorem:

    • Theorem Statement: If two angles are vertical angles, then they are congruent.

    • Relationship: Vertical Angles →\rightarrow Congruent.

  • Complement Theorem:

    • Theorem Statement: If two angles form a right angle, then they are complementary.

    • Relationship: Form a Right Angle →\rightarrow Complementary.

  • Linear Pair Theorem (Supplement Theorem):

    • Theorem Statement: If two angles form a linear pair, then they are supplementary.

    • Relationship: Form a Linear Pair →\rightarrow Supplementary.

  • Congruent Complements Theorem:

    • Theorem Statement: If two angles are complementary to the same angle (or to congruent angles), then they are congruent.

    • Formal Example: If ∠A\angle A is complementary to ∠B\angle B and ∠C\angle C is complementary to ∠B\angle B, then ∠A≅∠C\angle A \cong \angle C (or m∠A=m∠Cm\angle A = m\angle C).

  • Congruent Supplements Theorem:

    • Theorem Statement: If two angles are supplementary to the same angle (or to congruent angles), then they are congruent.

    • Formal Example: If ∠A\angle A is supplementary to ∠B\angle B and ∠C\angle C is supplementary to ∠B\angle B, then ∠A≅∠C\angle A \cong \angle C (or m∠A=m∠Cm\angle A = m\angle C).

Structure and Application of Two-Column Proofs

  • Two-Column Proof Overview:

    • A two-column proof is a standard format used to logically organize geometric and algebraic proofs.

    • Left Column: Contains the sequential statements (or steps).

    • Right Column: Contains the reasons justifying each individual statement.

    • Valid Reasons: Properties, definitions, postulates, and theorems.

  • Algebraic Two-Column Proof 1:

    • Given: 4x−1=274x - 1 = 27

    • Prove: x=7x = 7

    • Step 1:

      • Statement: 4x−1=274x - 1 = 27

      • Reason: Given

    • Step 2:

      • Statement: 4x=284x = 28

      • Reason: Addition property

    • Step 3:

      • Statement: x=7x = 7

      • Reason: Division property

  • Algebraic Two-Column Proof 2:

    • Given: a−6+2=5\frac{a}{-6} + 2 = 5

    • Prove: a=−18a = -18

    • Step 1:

      • Statement: a−6+2=5\frac{a}{-6} + 2 = 5

      • Reason: Given

    • Step 2:

      • Statement: a−6=3\frac{a}{-6} = 3

      • Reason: Subtraction property

    • Step 3:

      • Statement: a=−18a = -18

      • Reason: Multiplication property

  • Algebraic Two-Column Proof 3:

    • Given: −9(2x−3)=63-9(2x - 3) = 63

    • Prove: x=−2x = -2

    • Step 1:

      • Statement: −9(2x−3)=63-9(2x - 3) = 63

      • Reason: Given

    • Step 2:

      • Statement: −18x+27=63-18x + 27 = 63

      • Reason: Distributive Property

    • Step 3:

      • Statement: −18x=36-18x = 36

      • Reason: Subtraction property

    • Step 4:

      • Statement: x=−2x = -2

      • Reason: Division property

Property Identification Practice Examples

  • Example 1:

    • Statement: If k=3k = 3, then 3=k3 = k

    • Justifying Property: Symmetric Property

  • Example 2:

    • Statement: If 2x=142x = 14, then x=7x = 7

    • Justifying Property: Division Property

  • Example 3:

    • Statement: 10y=10y10y = 10y

    • Justifying Property: Reflexive Property

  • Example 4:

    • Statement: If −5x−1=−11-5x - 1 = -11, then −5x=−10-5x = -10

    • Justifying Property: Addition Property

  • Example 5:

    • Statement: If 10a=2b10a = 2b and 2b=c2b = c, then 10a=c10a = c

    • Justifying Property: Transitive Property

  • Example 6:

    • Statement: −7(n−4)=−7n+28-7(n - 4) = -7n + 28

    • Justifying Property: Distributive Property

  • Example 7:

    • Statement: If 6y=246y = 24, then 6y−3=24−36y - 3 = 24 - 3

    • Justifying Property: Subtraction Property

  • Example 8:

    • Statement: If 10x+w=4110x + w = 41 and w=1w = 1, then 10x+1=4110x + 1 = 41

    • Justifying Property: Substitution Property

  • Example 9:

    • Statement: If 3x=2y3x = 2y and 2y=z2y = z, then 3x=z3x = z

    • Justifying Property: Transitive Property

  • Example 10:

    • Statement: If 7m=357m = 35, then 7m+4=35+47m + 4 = 35 + 4

    • Justifying Property: Addition Property

  • Example 11:

    • Statement: If −2c=18-2c = 18, then 18=−2c18 = -2c

    • Justifying Property: Symmetric Property

  • Example 12:

    • Statement: Given 3x2+13x^2 + 1, if x=5x = 5, then 3(5)2+13(5)^2 + 1

    • Justifying Property: Substitution Property

  • Example 13:

    • Statement: If m=−2m = -2, then 8m=−168m = -16

    • Justifying Property: Multiplication Property

  • Example 14:

    • Statement: 5x+8x=x(5+8)5x + 8x = x(5 + 8)

    • Justifying Property: Distributive Property