Properties from AlgebraSection 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 inequalityReflexive Property

◦a = a (reflection in the mirror)Symmetric Property

◦If a = b, then b = aTransitive 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

Reflexive Property

Symmetric Property◦If ◦If

Transitive◦If ◦If F

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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◦ http://www.youtube.com/watch?v=8GWI0A9o_5E

Homework

Properties from Algebra worksheet #1-13 all

Vertical Angle Theorem◦Vertical angles are congruent.

1 2 3

4

∠𝟏 𝒂𝒏𝒅 ∠𝟐 𝒂𝒓𝒆 𝒗𝒆𝒓𝒕𝒊𝒄𝒂𝒍 𝒂𝒏𝒈𝒍𝒆𝒔. ∠𝟑 𝒂𝒏𝒅 ∠𝟒 𝒂𝒓𝒆 𝒗𝒆𝒓𝒕𝒊𝒄𝒂𝒍 𝒂𝒏𝒈𝒍𝒆𝒔.

Using the Vertical Angles Thm

Solve for x and find the measure of all the angles.x = 10, 140

1 23

4

Prove that vertical angles are congruentGiven:

Prove:

Statements Reasons

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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

http://www.nbcnews.com/video/nightly-news/52469048#52469048

http://video.msnbc.msn.com/the-cycle/49665334#49665334

Writing a 2-Column Proof

Problem 3 on p.116

Statements Reasons

Homework

p. 117 #5-13 oddp. 124 #6, 8, 9, 11, 12, 17p. 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 ABGiven: M is the midpoint of ABProve: 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

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Proving Theorems

Angle Bisector Theorem◦ If BX is the bisector of , then

m = ½ m and m = ½ mGiven: BX is the bisector ofProve: 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 m Substitution

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