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Common Notions. Axioms. Students will be able to identify the axioms used to solve algebraic equations Students will be able to solve geometric problems Axioms: a statement assumed to be true without proof. Objectives. Things that are equal to the same thing are equal to each other.
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Common Notions Axioms
Students will be able to identify the axioms used to solve algebraic equations Students will be able to solve geometric problems Axioms: a statement assumed to be true without proof. Objectives
Things that are equal to the same thing are equal to each other. • In Algebraic terms: • If a = b • b = c • Then a = c • You might think of this as substituting c for b Axiom 1
If equals are added to equals, the sums are equal. • In Algebra Terms: • If a = b and c = d • Then a + c = b + d Axiom 2
If equals are subtracted from equals, the differences are equal. • In Algebra Terms: • If a = b and c = d • Then a - c = b - d Axiom 3
Things that are alike or coincide with each other are equal to one another. • In Algebra Terms: • Two figures that can be made to match each other, point for point are equal. Axiom 4
The whole, or sum, is greater than the parts. • In Algebra Terms: • If a + b = c • a & b > 0 • Then c > a & c > b Axiom 5
Given: x – 3 = 10 +3 +3 x = 13 • Reason: • Axiom 2: If equals are added to equals, then the sums are equal. Example 1
Given: x + 3 = 10 -3 -3 x = 7 • Reason: • Axiom 3: If equals are subtracted from equals, then the differences are equal. Example 2
Given: 1 yard = 3 feet 1 foot = 12 inches 1 yard = 3(12 inches) = 36 inches • Reason: • Axiom 1: Things that are equal to the same thing are equal to each other. Example 3
Given: The upper half circle, called a semicircle, completely matches the lower half. (Think of folding the together along the center line or diameter.) • Therefore, the semicircles are equal. • Reason: • Axiom 4: Things that are alike or coincide with each other are equal to one another. Example 4
Given: m ABD + m DBC = 180o • Therefore: 180o > ABD and 180o > DBC • Reason: • Axiom 5: The whole, or sum, is greater than the parts. Example 5