Geometry

1. Geometry Day 13

2. Today’s Agenda • More practice with proofs

3. 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 • Transitive Property of Equality • Substitution Property of Equality • Distributive Property of Multiplication over Addition & Subtraction

4. 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.

5. 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

6. 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)

7. 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.

8. 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.

9. 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.

10. 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.

11. Homework • Homework 8 • Workbook, p. 25 • Homework 9 • Workbook, pp. 27, 30