some properties from algebra applied to geometry propertysegmentsangles reflexive symmetric...
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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