Properties from Algebra Section 2-5 p. 113

Properties of Equality Addition Property ◦If a = b and c = d, then a + c = b + d Subtraction Property ◦If a = b and c = d, then a – c = b – d Multiplication Property ◦If a = b then ca = cb Division Property ◦If a = b and c ≠ 0, then a/c = b/c

Properties of Equality (continued) Substitution Property ◦If a = b, then either a or b may be substituted for the other in any equation or inequality Reflexive Property ◦a = a (reflection in the mirror) Symmetric Property ◦If a = b, then b = a Transitive Property ◦If a = b and b = c, then a = c

Distributive Property a(b+c) = ab + ac a(b-c) = ab - ac

State the property used in each step above Problem 1 on p.114

Properties of Congruence __

Definitions (review) Congruent ◦Have equal measure Supplementary angles ◦Two angles whose measures have a sum of 180 Complementary angles ◦Two angles whose measures have a sum of 90 Vertical angles ◦Two angles whose sides are opposite rays Linear pair ◦Pair of adjacent angles whose non-common sides are opposite rays Angle bisector ◦Ray that divides an angle into two congruent angles Midpoint ◦Point that divides a segment into two congruent segments Segment bisector ◦Intersects a segment at its midpoint

More Definitions Perpendicular lines ◦Two lines that intersect to form right angles Perpendicular bisector ◦Is perpendicular to a segment at its midpoint

Segment Addition Postulate ◦If A, B, and C are collinear, and point B lies between points A and C, then AB+BC=AC Angle Addition Postulate ◦If point B lies on the interior of <AOC then m<AOB + m<BOC = m<AOC ◦Video ◦

Homework Properties from Algebra worksheet #1-13 all

Vertical Angle Theorem ◦Vertical angles are congruent.

Using the Vertical Angles Thm

StatementsReasons 1 3 2

More Theorems Congruent Supplements Theorem ◦If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent Congruent Complements Theorem ◦If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent All right angles are congruent. If two angles are congruent and supplementary, then each is a right angle

Proofs Deductive Reasoning (logical reasoning) ◦Process of reasoning logically from given statements or facts to a conclusion ◦If p  q is True ◦And p is True ◦Then q is true ◦Example: If a student gets an A on the final exam, then the student will pass the course. ◦Megan got an A on the final exam. What can you conclude? Proof ◦convincing argument that uses deductive reasoning

news/ # news/ # cycle/ # cycle/ #

Writing a 2-Column Proof Problem 3 on p.116 StatementsReasons

Homework p. 117 #5-13 odd p. 124 #6, 8, 9, 11, 12, 17 p. 124 #9

Planning a Proof Parts of a Proof ◦Statement of the Theorem (conditional statement; typically If-then statement) ◦Diagram showing given information ◦List of what is Given ◦List of what you are trying to Prove ◦Series of Statements and Reasons (lead from given information to the statement you are proving) ◦Remember that postulates are accepted without proof, but you have to prove theorems using definitions, postulates, and given information

Planning a Proof- Method 1 Gather as much info as you can. Reread what is given. What does it tell you? Look at the diagram. What other info can you conclude? Develop a plan to get from a to b (what you are given to what you are trying to prove).

Planning a Proof- Method 2 Work backward. Start with the conclusion (what you are trying to prove) Answer the question: This statement would be true if ________? Continue back to the Given statement, continuing to ask the same question: This statement would be true if ________? This becomes the plan for your proof.

Proving Theorems Midpoint Theorem ◦If M is the midpoint of AB, then AM = ½ AB and MB = ½ AB Given: M is the midpoint of AB Prove: AM = ½ AB; MB = ½ AB ◦Statements Reasons ◦M is the midpoint of AB Given ◦AM = MB Definition of Midpoint ◦AM + MB = AB Segment Addition Postulate ◦AM + AM = AB Substitution ◦MB + MB = AB Substitution ◦AM = ½ AB Division Property of Equality ◦MB = ½ AB Division Property of Equality __

Proving Theorems Angle Bisector Theorem ◦If BX is the bisector of, then m = ½ m and m = ½ m Given: BX is the bisector of Prove: m = ½ m and m = ½ m ◦Statements Reasons ◦BX is the bisector of Given ◦m = m Def of Angle Bisector ◦m + m = m Angle Add. Postulate ◦m + m = m Substitution ◦Or 2*m = m Substitution ◦m = ½ m Mult.or Div. Prop. of Equality ◦m = ½ mSubstitution → → →