Reasoning and Proof: An Introduction

1. REASONING AND
PROOF
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2. REASONING AND PROOF
Most geometric concepts expressed in the if-then clause are called
conditional statement. As discussed in the previous section, a
conditional is a statement that is formed by combining two words, if
and then.
Examples:
1. If your younger brother is six years old, then he can be
admitted to kindergarten.
2. If , then a = 2.
3. If you use this book and study it, then you will attain mastery of
grade 8 mathematical concepts.
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3. REASONING AND PROOF
There are two ways in writing a proof: through a two-column proof
and a paragraph proof.
Likewise, there are also two techniques in presenting a proof: direct
proving and indirect proving. The direct proof is usually written in a
two-column structure, while the indirect proof is often written in
paragraph form.
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5. REASONING AND PROOF
The most common step in writing a proof listed below.
1. Draw an accurate figure about what is to be proven.
2. Mark the figure according to what you can deduce about it
from the given information.
3. Write logical statements with corresponding reasons or
justifications. Corresponding reasons and statements must be
placed in proper order.
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7. ALGEBRAIC PROPERTIES OF
EQUALITY
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Let a, b, and c be real numbers.
Addition Property If a=b, then a+c=b+c.
Subtraction Property If a=b, then a-c=b-c
Multiplication Property If a=b, then ac=bc.
Division Property If a=b and c≠0, then a÷c=b÷c.
Reflexive Property For any real number a, a=a.
Symmetric Property If a=b, then b=a.
Transitive Property If a=b, and b=c, then a=c.
Substitution Property If a=b, then a can be substituted for b in
any equation or expression.

8. Linear Pair Theorem
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A linear pair consists of two adjacent angles whose noncommon sides
are opposite rays. Linear pairs add to 180°.

9. Complementary Angles
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Complementary Angles are two angles whose measures have the sum
of 90°.

10. Supplementary Angles
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Supplementary Angles are two angles whose measures have the sum
of 180°.

11. Segment Addition Postulate
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If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is
between A and C.

14. Examples:
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15. If 2x+3=13, then x=5
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Statement Reason
2x+3 = 13-3 Given
2x=10 SPE
x=5 DPE
Given: 2x+3 = 13
Prove: x = 5

16. If 2x+3=13, then x=5
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Statement Reason
2x + 3 = 13 Given
2x + 3 - 3 = 13 - 3 SPE
2x = 10 Simplification/ Subtraction
2x/2 = 10/2 DPE
x = 5 Simplification/ Division
Given: 2x+3 = 12
Prove: x = 5

17. If 3(x+5)-x=29, then x=7
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Statement Reason
3(x+5)-x=29 Given
3x+15-x=29 Distributive PoE
2x+15=29-15 Combine Like Terms
2x=14 SPE
x=7 DPE
Given: 3(x+5)-x = 29
Prove: x = 7

18. If 3(x+5)-x=29, then x=7
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Statement Reason
3(x+5)-x = 29 Given
3x+15-x = 29 Distributive PoE
3x-x+15-15=29-15 SPE/Combining Like Terms
2x=14 Simplification
2x/2=14/2 DPE
x = 7 Simplification
Given: 3(x+5)-x = 29
Prove: x = 7

19. Example 2:
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Statement Reason
RT=RS+ST Segment Add. p.
5x-12=x+2 + 3x-8 Substitution
5x-12=4x-6+12 Combine Like Terms
5x=4x+6 APE
x=6 SPE
Use a postulate or theorem to find
the value of x in each figure.
Given: RT = 5x - 12

20. Example 2:
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Statement Reason
RT = RS + ST Segment Addition Postulate
5x-12 = (x+2) + (3x - 8) Substitute
5x-12 = 4x - 6 Simplification/Combine Like Terms
x-12 = -6 SPE
x = 6 APE
Use a postulate or theorem to find
the value of x in each figure.
Given: RT = 5x - 12

21. Example 3:
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Statement Reason
m<RST = m<RSP + m<PST Angle Add. p
15x-10=x+25 + 5x+10 Substitution
15x-10=6x+35 Combine Like Termd
Use a postulate or theorem to find
the value of x in each figure.
Given: m∠RST = (15x-10)°

22. Example 3:
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Statement Reason
m∠RST = m∠RSP + m∠PST Angle Addition Postulate
(15x-10)° = (x+25)° + (5x+10)° Substitution
15x-10=6x+35 Combine Like Terms/simplication
9x-10=35 SPE
9x = 45 APE
x = 5 DPE
Use a postulate or theorem to find
the value of x in each figure.
Given: m∠RST = (15x-10)°