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Math 1312
Section 1.5
Introduction to Geometric Proof
Properties of Equality (Table 1.5) page 40
Addition Property of Equality
If a = b , then a + c = b + c
Subtraction Property of Equality
If a = b , then a − c = b − c
Multiplication Property of Equality
If a = b , then a × c = b × c
a b
If a = b and c ≠ 0 , then =
c c
Division Property of Equality
Example 1: Which property justifies the conclusion of the statement?
1. If 3 x = 9 , then x = 3 .
2. If x + 2 = 10 , then x = 8 .
3. If
2
x = 8 , then x = 12 .
3
Further properties of Algebra (Table 1.6) page 40
Distributive Property
a (b + c) = a × b + a × c .
Substitution Property
If a = b , then a replaces b in any equation.
Transitive Property
If a = b and b = c , then a = c .
Symmetric Property
If a = b , then b = a .
Reflexive Property
a=a
Example 2: Given: 3 x + 2 = 4 + 5 x
Prove: x = −1
PROOF
Statements
1.
Reasons
3x + 2 = 4 + 5 x
2. 3 x + 2 − 4 = 4 + 5 x − 4
3. 3 x − 2 = 5 x
4. 3 x − 3 x − 2 = 5 x − 3 x
5. − 2 = 2 x
6.
1
1
( − 2) =   2 x
2
2
7. −1 = x
8. x = −1
Example 3: Given: B is the midpoint of the line segment AC
AC
Prove: AB =
2
PROOF
Statements
1. B is the midpoint of AC
2. AB = BC
3. AB + BC = AC
4. AB + AB = AC
5. 2 AB = AC
6. AB =
AC
2
Study the examples in the book for this section!
Reasons
M 1312
Section 1.5
Example 3: State the conclusion AND the property that justifies it.
1. If m∠1 + m∠2 = 900 and m∠3 = m∠1 ; what is true?
2. K is in the interior of ∠GHJ ; what can we conclude about m∠GHK + m∠KHJ ?
3. Suppose that m∠ABC = 1280 . If BD bisects ∠ABC , determine m∠ABD .
4. m∠1 + m∠2 = 1800
3