Transcript s - Angelfire

Some properties from algebra
applied to geometry
Property
Segments
Angles
Reflexive
PQ=QP
Symmetric
If AB= CD,
then CD = AB.
If m<A = m<B,
then m<B = m<A
Transitive
If GH = JK and
JK = LM, then
GH = LM
If m<1 = m<2 and
m<2 = m<3, then
m<1 = m<3
m<1 = m<1
Examples
Name the property of equality that justifies each statement.
Statement
Reasons
If AB + BC=DE + BC, then AB = DE
Subtraction property (=)
m<ABC= m<ABC
Reflexive property (=)
If XY = PQ and XY = RS,
then PQ = RS
If (1/3)x = 5, then x = 15
If 2x = 9, then x = 9/2
Substitution property (=)
Multiplication property (=)
Division property (=)
Example 2
Justify each step in solving 3x + 5 = 7
2
Statement
The previous
3x + 5 = 7
example is a
2
proof of the
2(3x + 5) = (7)2 conditional:
2
If 3x + 5 = 7,
2
3x + 5 = 14
then x=3
Reasons
Given
Multiplication property (=)
Distributive property (=)
3x = 9
Subtraction property (=)
x=3
Division property (=)
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  AB
Transitive property:
If AB  CD, and CD  EF, then AB  EF
Abbreviation: reflexive prop. of  segments
symmetric prop. of  segments
transitive 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  <s
symmetric prop. of  <s
transitive 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