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Chapter 2 By: Nick Holliday Josh Vincz, James Collins, and Greg “Darth” Rader

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Section 1 Vocab ● Conditional statement- statement with a hypothesis and a conclusion ● Hypothesis- “if ” part of a conditional statement. ● Conclusion- “then” part of the conditional statement ● If then form- if contains the hypothesis and then contains the conclusion ● Converse- The statement formed by switching the conclusion and the hypothesis ● Negation- The negative of a statement ● Inverse- The statement formed when you negate the hypothesis and conclusion of the converse.

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Section 1 Vocab Continued ● Contrapositive- The statement formed when you negate the hypothesis and conclusion of a conditional statement. ● Equivalent statement- 2 statements that are both true or that are both false.

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Example 1 ● Rewrite the conditional statement. ● An even number is divisible by 2 ● Conditional statement = If it is an even number Then it is divisible by 2

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Example 2 ● Write the (a inverse, (b converse, (c contrapositive of the following statement. ● If it is Friday then there is no school tomorrow. ● (a Inverse: If it is not Friday, then there is school tomorrow. ● (b Converse: If there is no school tomorrow, then it is Friday. ● (c Contrapositive: If there is school tomorrow, then it is not Friday.

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Checkpoint ● Write the inverse, converse, and contrapositive of the conditional statement. ● If Josh is complaining about a test score Then he was in Mrs. Wagner's class. ●

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Point line and plane postulate` ● Post 5: Through any two points there exists exactly one line ● Post 6: A line contains at least two points ● Post 7: If two lines intersect then their intersection is one point ● Post 8: Through any three non colinear points there exists one plane ● Post 9: A plane contains at least three noncolinear points ● Post 10: If two points lie in a plane, then the line containing them lies in the plane ● Post 11: If two planes intersect, then their intersection is a line

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Section 2 vocab ● Perpendicular lines – two lines that form a right angle. ● Line perpendicular to a plane- intersects plane at point that is perpendicular to every line. ● Bioconditional statement- a statement that contains if and only if and conditional and converse.

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Example ● If it is an equailateral triangle then all angles on the triangle are congruent. ● If all the angleson the triangle are congruent then it is an equalaterial triangle ● Since both statements are true the biconditional statement is... ● It is an equalateral triangle if and only if all of the angles on the triangle are congruent.

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Section 3 Vocab ● Logical argument- an argument based on deductive reasoning which uses facts, definintions, and accepted properties in a logical order ● Law of Detachment- If p q is a true conditional and p is true then q is true ● Law of Syllogism- If p and q r are true conditional statements, then p r is true

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Other notes of section 3 ● P hypothesis ● Q conclusion ● Conditional statement = p q ● Converse = q p ● Biconditional statement = p< q or ● P if and only if Q ● ~ negate that portion of the statement

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Example ● Let p be value of x is 7. Let q be x is <10. ● Write p—q in words then write q—p in words. ● Decide whether the Biconditional statement p<>q is true.

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Algebraic properties of equality ● Let a b and c be real numbers. ● Addition property- if a= b then a+c=b+c ● Subtraction property- if a=b then a-c=a-b Multiplication property- if a=b then ac=bc ● Division property- if a=b and c does not = c then a/c=b/c ● Reflexive property- for any real number a, a=a ● Symmetric property- if a=b then b=a ● Transitive property- if a=b and b=c then a=c ● Substitution property- if a=b, then a can be substituted for b in any equation

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Properties of Equality ● Segment Length ● Reflexive- For any segment AB AB=AB ● Symmetric- If AB=CD then CD=AB ● Transitive- If AB=CD and CD=EF then AB=EF ● Angle Measure ● Reflexive- For any angle A m<A =m<A ● Symmetric- If m<A=m<B then m<B=m<A ● Transitive- If m<A=m<B and m<B=m<C, then m<A=m<C

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Example ● Solve the following equations -2x +1 =56 -3x 5x + 12 = 2 + 10x

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Section 5 vocab Theorem- A true statement that follows as a result of other true statements Two-column Proof- A type of proof written as numbered statement and reasons that show a logical argument Paragraph Proof- type of proof written as a paragraph.

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Theorem 2.1 ● Reflexive- for any segment ab, ab is congruent to ab. ● Symmetric- if ab is congruent to cd then cd is congruent to ab. ● Transitive- if ab is congruent to cd and cd is congruent to ef then ab is congruent to ef.

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Example ● Given JK is congruent to MN. MN is congruent to PQ. Prove JK is congruent to PQ

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Section 2.6 ● Theorem 2.2 properties of angle congruences. ● Reflexive- for any angle a, a=a ● Symmetric- if angle a is congruent to angle b then angle b is congruent of angle a. ● Transitive- if angle a is congruent to angle b and angle b is congruent to angle c then angle a is congruent angle c.

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Example ● Given that angle 4 is congruent to angle 6 and angle 6 is congruent to angle 8. The measure of angle 8 is 77. what is the measure of angle 4. explain your reasoning.

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Theorem 2.3+ theorem 2.4 ● All right angles are congruent. ● If two angles are supplementary then they are congruent. ● If angle 1 + angle 2 = 180 and angle 2+ angle 3 = 180 then angle 1 and angle 3 are congruent.

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Theorem 2.5 ● If two angles are complementary to the same angle then the two angles are congruent ● If angle 4 + angle 5=90 and angle 5+angle 6=90 then angle 4 = angle 6

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Example ● Given angle 1 and angle 2 are complements, angle 3 and angle 4 are complements, angle 2 and angle 4 are congruent. Prove angle 1 and angle 3 are congruent,

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Chapter 2 Reasoning and Proof.

Chapter 2 Reasoning and Proof.

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