Geometry

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Geometry. Day 13. Today’s Agenda. More practice with proofs. Properties of Real Numbers (p. 134). Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Reflexive Property of Equality Symmetric Property of Equality

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Geometry

Day 13

Today’s Agenda
• More practice with proofs
Properties of Real Numbers (p. 134)
• Subtraction Property of Equality
• Multiplication Property of Equality
• Division Property of Equality
• Reflexive Property of Equality
• Symmetric Property of Equality
• Transitive Property of Equality
• Substitution Property of Equality
• Distributive Property of Multiplication over Addition & Subtraction
Algebraic Proof
• An algebraic proof is a proof made up of a series of algebraic statements, each justified by a property of real numbers.
• Let’s review Example 1 on p. 134.
• Algebraic proofs will be good practice for Geometric proofs, because the format is similar but you are already familiar with the concepts involved.
• Let’s look at the format:
• There are two columns. The left column is a step-by-step process that solves the problem. The right column contains the justification for each statement on the left side.
• This format is called a two-column, or formal proof.
Geometric Proofs
• Proofs always start with the provided, or given information. (Deductive reasoning must have a starting premise.) They will always end with the conjecture we are trying to prove having been demonstrated.
• Let’s examine Example 3 on p. 136, and then complete the Guided Practice problems.
• Proofs can be a difficult concept to master. For further information of proofs:http://mathforum.org/dr.math/faq/faq.proof.html
Geometric Proofs
• A geometric proof will contain:
• The given information
• The conjecture (what you’re trying to prove)
• A labeled diagram
• Statements (left-hand column)
• Reasons (right-hand column)
Line Segment Postulates (p. 142)
• Ruler Postulate – The points on any line or line segment can be put into one-to-one correspondence with real numbers.
• Segment Addition Postulate – If A, B, and C are collinear, then point B is between A and C if and only if AB + BC = AC.
• Let’s go through Example 1 on p. 143. When we are attempting to prove that segments are congruent, the idea is that we will show their measures are the same.
• Form groups of 3-4, and complete Guided Practice 1.
Line Segment Theorems (p. 143)
• Some of the Algebraic Properties also apply to Geometric figures:
• Reflexive Property of Congruence
• Symmetric Property of Congruence
• Transitive Property of Congruence
• We’ll prove these each of these.
• Now each group will be assigned a problem from 9-12 on p. 146. This will be turned in for an assignment grade, along with another proof yet to come.
Angle Postulates (p. 149)
• Protractor Postulate – Given any angle, the measure can be put into one-to-one correspondence with real numbers between 0 and 180.
• Angle Addition Postulate – D is the interior of ABC if and only if mABD + mDBC = mABC.
• Let’s go through Example 1 and Guided Practice 1 on p. 149.
Angle Theorems (p. 150-153)
• Supplement Theorem – If two angles form a linear pair, then they are supplementary angles.
• Complement Theorem – If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles.
• Properties of Angle Congruence:
• Reflexive Property of Congruence
• Symmetric Property of Congruence
• Transitive Property of Congruence
• Congruent Supplements Theorem – Angles supplementary to the same angle or to congruent angles are congruent.
• Congruent Complements Theorem – Angles complementary to the same angle or to congruent angles are congruent.
• Vertical Angles Theorem – If two angles are vertical angles, then they are congruent.
• Right Angle Theorems:
• Perpendicular lines intersect to form four right angles.
• All right angles are congruent.
• Perpendicular lines form congruent adjacent angles.
• If two angles are congruent and supplementary, then each angle is a right angle.
• If two congruent angles form a linear pair, then they are right angles.
• Groups will prove italicized theorems.
Homework
• Homework 8
• Workbook, p. 25
• Homework 9
• Workbook, pp. 27, 30