advanced geometry section 2.5 and 2.6
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Advanced Geometry Section 2.5 and 2.6. Addition, Subtraction, Multiplication, and Division Properties. Addition, Subtraction, Multiplication and Division Properties. I CAN… Use the addition, subtraction, multiplication and division properties - PowerPoint PPT PresentationTRANSCRIPT
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ADVANCED GEOMETRY SECTION 2.5 AND 2.6Addition, Subtraction, Multiplication, and
Division Properties
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Addition, Subtraction, Multiplication and Division Properties
I CAN…• Use the addition, subtraction,
multiplication and division properties
• Write proofs involving the addition, subtraction, multiplication and division properties
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Quick Review Define
complementary angles
Define supplementary angles
Define congruent segments
Define congruent angles
Two angles are complementary if their sum is 90°
Two angles are supplementary if their sum is 180°
Two segments are congruent if their measures are equal.
Two angles are congruent if they have the same measure.
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TheoremsIf a segment is added to two congruent segments, the sums are congruent. (Addition Property)
7cm3cm7 cmA B C D
AB + BC = CD + BC
AC = BD, so AC BD
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TheoremsIf an angle is added to two congruent angles, the sums are congruent (Addition Property)
Note that we first need to know that we have 2 congruent angles, then that we are adding the same angle to both
mABC + mCBD = mDBE + mCBD
mABD = mCBE, so mDBE = 50.03
mABC = 50.03A
B
E
C
D
ABD CBE
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TheoremsIf congruent segments are added to congruent segments, the sums are congruent. (Addition Property)
CF + FG = HE + EDD E
GC
H
F
CG = HD , so
CG HD
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TheoremsIf congruent angles are added to congruent angles, the sums are congruent. (Addition Property)
mJIL + mLIK = mJKL + mLKIJIK JKI
J
I K
L
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TheoremsIf a segment (or angle) is subtracted from congruent segments (or angles), the differences are congruent. (Subtraction Property)
10
10
Q RB A
QR - BR = BA - BR
QB BR
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TheoremsIf a segment (or angle) is subtracted from congruent segments (or angles), the differences are congruent. (Subtraction Property)
mABD - mCBD = mCBE - mCBDmABD = 78
mCBE = 78D
C
E
B
A
ABC DBE
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Using the Addition and Subtraction Properties
An addition property is used when the segments or angles in the conclusion are greater than those in the given information
A subtraction property is used when the segments or angles in the conclusion are smaller than those in the given information.
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Given:
Conclusion:
GJ HK
GH JK G K
M
H J
Statements Reasons1. 1. Given
2. 2. Subtraction
GJ HK
GH JK
How to use this theorem in a proof:
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Multiplication Property If segments (or angles) are congruent,
then their like multiples are congruent.
If B, C, F, and G are trisection points and then by the Multiplication Property.
A B C D E F G H
,EFABEHAD
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Division Property If segments (or angles) are congruent, then their
like divisions are congruent.
If are angle bisectors, then by the division property.
C
SA
T
D
ZO
G
OZandASandDOGCAT ,DOZCAS
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Using the Multiplication and Division Properties in Proofs
Look for a double use of the word midpoint, trisects, or bisects in the “Given.”
Use multiplication if what is Given < the Conclusion
Use division if what is Given > the Conclusion
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Example Given:
O is the midpoint of R is the midpoint of
Prove: Statements Reasons
NSMPMPNS
NRMO
M O P
N R S
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More Examples and Homework Read Sample Problems 2 through 4 on
pages 90 and 91. HW: p. 86 #4-6, 11; p. 91 #1, 3, 4, 11, 12
Don’t forget to draw all the diagrams!!!!!