Some properties from algebra applied to geometry

Property Segments Angles

Reflexive

Symmetric

Transitive

PQ=QP m<1 = m<1

If AB= CD, then CD = AB.

If m<A = m<B, then m<B = m<A

If GH = JK and JK = LM, then GH = LM

If m<1 = m<2 and m<2 = m<3, then m<1 = m<3

ExamplesName the property of equality that justifies each statement.

Statement Reasons

If AB + BC=DE + BC, then AB = DE

m<ABC= m<ABC

If XY = PQ and XY = RS,

then PQ = RS

If (1/3)x = 5, then x = 15

If 2x = 9, then x = 9/2

Subtraction property (=)

Reflexive property (=)

Substitution property (=)

Multiplication property (=)

Division property (=)

Example 2Justify each step in solving 3x + 5 = 7

2Statement Reasons

Given

Multiplication property (=)

Distributive property (=)

Subtraction property (=)

Division property (=)

3x + 5 = 7

22(3x + 5) = (7)2

23x + 5 = 14

3x = 9

x = 3

The previous example is a proof of the conditional:

If 3x + 5 = 7,

2then x=3

This type of proof is called a TWO-COLUMN PROOF

Verifying Segment Relationships

Five essential parts of a good proof:•State the theorem to be proved.•List the given information.•If possible, draw a diagram to illustrate the given information.•State what is to be proved.•Develop a system of deductive reasoning. (Use definitions, properties, postulates, undefined terms, or other theorems previously proved).

Theorem 2.1 Congruence of segments is reflexive, symmetric, and transitive. Reflexive property: AB AB.Symmetric property: If AB CD, then CD ABTransitive property:

If AB CD, and CD EF, then AB EF

Abbreviation: reflexive prop. of segmentssymmetric prop. of segmentstransitive prop. of segments

Verifying Angle Relationships

Theorem 2-2 Supplement Theorem: If two angles form a linear pair, then they are supplementary angles

Theorem 2-3: Congruence of angles is reflexive, symmetric and transitive.

Abbreviation: reflexive prop. of <ssymmetric prop. of <stransitive prop. of <s

Theorem 2-4: Angles supplementary to the same angle or to congruent angles are congruent:

Abbreviation: <s supp. to same < or <s are

Verifying Angle Relationships

Theorem 2-5: Angles complementary to the same angle or to congruent angles are congruent

Abbreviation: <s comp. to same < or <s are Theorem 2-6 : All right angles are congruent

Abbreviation: All rt. <s are

Theorem 2-7: Vertical angles are congruent. Abbreviation: Vert. <s are

Theorem 2-8: Perpendicular lines intersect to form four right angles.

Abbreviation: lines form 4 rt. <s