Download Reasoning Using Algebra Properties

Lesson 2
COMMON CORE MATHEMATICS CURRICULUM
U2
GEOMETRY
Name_________________________ Date_________
2.5 Reasoning Using Algebra Properties
Goal.  Use algebraic properties in logical arguments.
ALGEBRAIC PROPERTIES OF EQUALITY
Let 𝑎, 𝑏, and 𝑐 be real numbers.
Addition Property
If 𝑎 = 𝑏, then _______________________________.
Subtraction Property
If 𝑎 = 𝑏, then _______________________________.
Multiplication Property
If 𝑎 = 𝑏, then _______________________________.
Division Property
If 𝑎 = 𝑏 and 𝑐  0. then_______________________.
Substitution Property
If 𝑎 = 𝑏, then _______________________________
____________________________________________.
DISTRIBUTIVE PROPERTY
a(b + c) =____________________________, where a, b, and c are real numbers.
Show how to use algebraic properties in preparation for two-column
proofs:
1. Solve 8𝑥 – 5 = – 2𝑥 – 15. Write an explanation (what you did) and reason for
each step – algebraic property if defined.
Solution
Equation
Explanation
Reason
A. 8𝑥 – 5 = – 2𝑥 – 15
Original equation
Given
B. 10𝑥 – 5 = – 15
C. 10𝑥 = – 10
D. 𝑥 = – 1
2. Solve 𝟒(𝟔𝒙 + 𝟐) = 𝟔𝟒
Equation
A. 𝟒(𝟔𝒙 + 𝟐) = 𝟔𝟒
Explanation
Original equation
Reason
Given
B.
C.
D.
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COMMON CORE MATHEMATICS CURRICULUM
Lesson 2
U2
GEOMETRY
Reasoning Using RST Properties
REFLEXIVE PROPERTY OF EQUALITY (1)
Real Numbers
For any real number a, ______.
Segment Length
For any segment AB, __________.
Angle Measure
For any angle A, _________________.
SYMMETRIC PROPERTY OF EQUALITY (2)
Real Numbers
For any real numbers a and b, if a = b,
then ________.
Segment Length
For any segments AB and CD, if AB = CD,
then__________.
Angle Measure
For any angles A and B, and if mA =
mB then ________.
TRANSITIVE PROPERTY OF EQUALITY (3)
Real Numbers
For any real numbers a, b, and c, if a = b
and b= c, then ___________.
Segment Length
For any segments AB ,CD, and EF if AB =
CD ,and CD = EF, then__________.
Angle Measure
For any angles A, B, and C, if mA =
mB and mB = mC, then ________.
Chapter 1 Postulates:
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Lesson 2
COMMON CORE MATHEMATICS CURRICULUM
U2
GEOMETRY
Proofs using properties of equality
Proof 1: Show that CF = AD.
Equation
A. AB = _____
Look at the diagram and use the information.
Explanation
Marked in diagram
Reason
Given
B. BC = _____
C. AC = AB + BC
D. DF = _____ + _____
Segment Addition Postulate
E. DF = BC + AB
_____________ Property of Equality
F. DF = ______
_____________ Property of Equality
G. DF + CD = ________ + CD
_____________ Property of Equality
H. _____ = _____
Checkpoint Complete the following exercises. In Exercises 1-3, name the property
of equality that the statement illustrates.
1. If GH = JK, then JK = GH.
_______________________________________________________________________________
_______________________________________________________________________________
2. If r = s, and s = 44, then r = 44.
_______________________________________________________________________________
_______________________________________________________________________________
3. mN = mN
_______________________________________________________________________________
_______________________________________________________________________________
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Lesson 2
COMMON CORE MATHEMATICS CURRICULUM
U2
GEOMETRY
Use the property to complete the statement.
1. Addition Property of Equality: if RS = TU, then RS + 20 = _____________________.
2.
Multiplication Property of Equality: If m 1 = m 2, then 3m 1 = ____________.
3.
Substitution Property of Equality: If a = 20, then 5a = _______________________.
4.
Reflexive Property of Equality: If x is a real number, then x = __________________.
5.
Symmetric Property of Equality: If AB = CD, then CD = _______________________.
6.
Transitive Property of Equality: If m E = m F and m F = m G, then ________.
7.
Multiplication Property of Equality: If RS = TU, then x(RS) = ___________________.
8.
Division Property of Equality: If 3(m1) = m2, then m1 = _________________.
9.
Transitive Property of Equality: If ab = bc and bc = de, then ___________________.
10. Substitution Property of Equality: If x = 3c and r = 5x + 7, then ________________.
Proof 2:
Show that AC = 2(AB)
Equation
A. AB = BC
Explanation
Marked in diagram
Reason
Given
B. AC = AB + BC
C. AC = AB + AB
D. AC = 2(AB)
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Lesson 2
COMMON CORE MATHEMATICS CURRICULUM
U2
GEOMETRY
Proof 3: Show that mAEC = mBED
Equation
A. mAEB = mCED
Explanation
Marked in
diagram
Reason
Given
B. mBEC = mBEC
C. m AEB + mBEC = mCED + m
BEC
D. mAEC = mAEB + m BEC
E. mBED = mCED + m BEC
F. mAEC = mBED
Use the property to complete the statement.
1. Reflexive Property of Angle Measure: mB = __?__.
2.
Transitive Property of Equality: If CD = GH and = RS, then __?__.
3.
Addition Property of Equality: If x = 3, then 14 + x = __?__.
4.
Symmetric Property of Equality: If BC = RL, then __?__.
5.
Substitution Property of Equality: If mA = 45°, then 3(mA) = __?__.
6.
Multiplication Property of Equality: If mA = 45°, then __?__ (mA) = 15°.
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