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Geometry Journal 2

Geometry Journal 2. Nicolle Busto 9-1. 9-1. Conditional Statement. It is a statement that establishes a necessary condition for a thing to happen. Examples : If it rains then the grass will be wet . Converse: If the grass is wet then it rained .

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Geometry Journal 2

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  1. Geometry Journal 2 Nicolle Busto 9-1 9-1

  2. Conditional Statement • It is a statement that establishes a necessary condition for a thing to happen. Examples: • Ifitrainsthenthegrasswill be wet. • Converse: Ifthegrassiswetthenitrained. • Inverse: Ifitdoesn’t rain thenthegrasswillnot be wet. • Contrapositive: Ifthegrassisnotwetthenitdidn’t rain.

  3. Counter example • It’s an example that proves that a statement is false. Example: 1. If the grass is wet then it rained. -The grass could be wet, if you watered the grass. 2. If a number is divisible by 5 then its unit digit is 5. -20 3. If a number is odd then the number is prime. -9

  4. Definition A description of a mathematical concept that can be written as a bi-conditional statement. A line is perpendicular to a plane iff their intersection is exactly a 90° angle. Perpendicular lines: Two lines are perpendicular iff they intersect at 90°.

  5. Bi-Conditional statement It is a statement where both conditions are needed. They are used to write definitions They are important because they can be used to establish if a definition is true or false. Example 1: A rectangle is a square iff their four sides are equal. Example 2: An angle is acute iff its measure is less than 90°. Example 3: A triangle is equilateral iff its three sides are congruent.

  6. Deductive reasoning • Examples: p = today is a school day. Q = I am going to the school today. • - P  Q • P = todaythe ice creamstoreis open. Q = I am goingtothestoretoday. • P  Q • 3. P = Iftwonumbers are odd. Q = their sum iseven. Is the process of using logic to draw conclusions. Symbolic notation: Uses letters to represent statements and symbols to represent the connections between them. It works by reaching conclusions from a starting statement.

  7. Laws of Logic 1. Law of Detachment: If P  Q is a true statement and P is true then Q is true. ·Example 1: If an animal is a dog  it has a tail. - The pet of Busto family is a dog  the pet of Busto family has a tail. ·Example 2: If a number is even  it is divisible by two. - 120 is an even number  120 is divisible by two. ·Example 3: It rains today  the streets will be wet. - Tomorrow will rain  the streets will be wet tomorrow.

  8. 2. Law of Syllogism: If P  Q is true and Q  R is true, then P  R is true. ·Example 1: If an animal is a fish  it lives in water, if an animal lives in water  it is wet all the time. - A shark is a fish  a shark is wet all the time. ·Example 2: If an animal is a mammal  it has warm blood, if an animal has warm blood  it can control it’s inner temperature. - A dog is a mammal  it can control it’s inner temperature. ·Example 3: If a number is divisible by 12  it is divisible by 6, if a number is divisible by 6  it is even. - 36 is divisible by 12  36 is even.

  9. AlgebraicProof • You use theproperties and workwiththeinitialstatementuntilyoureachthedesiredconclusion.

  10. Properties of equality

  11. Properties of congruence

  12. Segments and Angles properties You can work with the measures of angles or measures of sides using the laws of algebra.

  13. P Q R S T U PQ = PQ PQ = RS, then RS = PQ If PQ = RS and RS = TU, then PQ = RS

  14. Proof process

  15. Two proof- column • You list your steps on the proof in the left column. You write the matching reason for each step in the right column. • They give you the given, and what you need to proof. Sometimes they give you the plan.

  16. Given: <1 and <2 are right angles Prove: <1 ~= <2 Plan: Use the definition of a right angle to write the measure of each angle. Then use the Transitive Property and the definition of congruent angles.

  17. Given: AB ~= XY BC ~= YZ Prove: AC ~= XZ

  18. Given: PQ ~= RS Prove: RS ~= PQ

  19. Linear pair postulate • Linear pair: a pair of adjacent angles whose no common sides are opposite rays. • LPP: If two angles form a linear pair, then they are supplementary angles. EXAMPLES: 1. 3. 2.

  20. Theorems

  21. Vertical anglestheorem • Vertical angles are congruent. 1. 2. 3.

  22. CommonSegmentTheorem • Give collinear points A, B, C, and D arranged as shown, if AB˜= CD, thenAC ˜= BD. • AB ˜= BD. • AC˜= BD. EXAMPLES: 1.

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