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Geometry Cliff Notes

Explore reasoning, proof techniques, lines, congruent triangles, and formulas in Geometry Cliff Notes Chapters 4 and 5.

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Geometry Cliff Notes

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  1. Geometry Cliff Notes Chapters 4 and 5

  2. Chapter 4 Reasoning and Proof, Lines, and Congruent Triangles

  3. Distance Formula d= Example: Find the distance between (3,8)(5,2) d=

  4. Midpoint Formula M= Example: Find the midpoint (20,5)(30,-5) M=

  5. Conjecture An unproven statement that is based on observations.

  6. Inductive Reasoning Used when you find a pattern in specific cases and then write a conjecture for the general case.

  7. Counterexample A specific case for which a conjecture is false. Conjecture: All odd numbers are prime. Counterexample: The number 9 is odd but it is a composite number, not a prime number.

  8. Conditional Statement A logical statement that has two parts, a hypothesis and a conclusion. Example: All sharks have a boneless skeleton. Hypothesis: All sharks Conclusion: A boneless skeleton

  9. If-Then Form A conditional statement rewritten. “If” part contains the hypothesis and the “then” part contains the conclusion. Original: All sharks have a boneless skeleton. If-then: If a fish is a shark, then it has a boneless skeleton. ** When you rewrite in if-then form, you may need to reword the hypothesis and conclusion.**

  10. Negation Opposite of the original statement. Original: All sharks have a boneless skeleton. Negation: Sharks do not have a boneless skeleton.

  11. Converse To write a converse, switch the hypothesis and conclusion of the conditional statement. Original: Basketball players are athletes. If-then: If you are a basketball player, then you are an athlete. Converse: If you are an athlete, then you are a basketball player.

  12. Inverse To write the inverse, negate both the hypothesis and conclusion. Original: Basketball players are athletes. If-then: If you are a basketball player, then you are an athlete. (True) Converse: If you are an athlete, then you are a basketball player. (False) Inverse: If you are not a basketball player, then you are not an athlete. (False)

  13. Contrapositive To write the contrapositive, first write the converse and then negate both the hypothesis and conclusion. Original: Basketball players are athletes. If-then: If you are a basketball player, then you are an athlete. (True) Converse: If you are an athlete, then you are a basketball player. (False) Inverse: If you are not a basketball player, then you are not an athlete. (False) Contrapositive: If you are not an athlete, then you are not a basketball player. (True)

  14. Equivalent Statement When two statements are both true or both false.

  15. Perpendicular Lines Two lines that intersect to form a right angle. Symbol:

  16. Biconditional Statement When a statement and its converse are both true, you can write them as a single biconditional statement. A statement that contains the phrase “if and only if”. Original: If a polygon is equilateral, then all of its sides are congruent. Converse: If all of the sides are congruent, then it is an equilateral polygon. Biconditional Statement: A polygon is equilateral if and only if all of its sides are congruent.

  17. Deductive Reasoning Uses facts, definitions, accepted properties, and the laws of logic to form a logical statement.

  18. Law of Detachment If the hypothesis of a true conditional statement is true, then the conclusion is also true. Original: If an angle measures less than 90°, then it is not obtuse. m <ABC = 80° <ABC is not obtuse

  19. Law of Syllogism If hypothesis p, then conclusion q. If hypothesis q, then conclusion r. (If both statements above are true). If hypothesis p, then conclusion r Original: If the power is off, then the fridge does not run. If the fridge does not run, then the food will spoil. Conditional Statement: If the power if off, then the food will spoil.

  20. Postulate A rule that is accepted without proof.

  21. Theorem A statement that can be proven.

  22. Subtraction Property of Equality Subtract a value from both sides of an equation. x +7 = 10 -7 -7 X = 3

  23. Addition Property of Equality Add a value to both sides of an equation. X-7 = 10 +7 +7 X = 17

  24. Division Property of Equality Divide both sides by a value. 3x = 9 • 3 x = 3

  25. Multiplication Property of Equality Multiply both sides by a value. ½x = 7 ·2 ·2 x = 14

  26. Distributive Property To multiply out the parts of an expression. 2(x-7) 2x - 14

  27. Substitution Property of Equality Replacing one expression with an equivalent expression. AB = 12, CD = 12 AB= CD

  28. Proof Logical argument that shows a statement is true.

  29. Two-column Proof Numbered statements and corresponding reasons that show an argument in a logical order.

  30. Reflexive Property of Equality Segment: For any segment AB, AB AB or AB = AB Angle: For any angle or

  31. Symmetric Property of Equality Segment: If AB CD then CD AB or AB = CD Angle:

  32. Transitive Property of Equality Segment: If AB CD and CD EF, then AB EF or AB=EF Angle:

  33. Supplementary Angles Two Angles are Supplementary if they add up to 180 degrees.

  34. Complementary Angles Two Angles are Complementary if they add up to 90 degrees (a Right Angle).

  35. Segment Addition Postulate If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C. . . . A B C

  36. Angle Addition Postulate If S is in the interior of angle PQR, then the measure of angle PQR is equal to the sum of the measures of angle PQS and angle SQR.

  37. Right Angles Congruence Theorem All right angles are congruent.

  38. Vertical AnglesCongruence Theorem Vertical angles are congruent.

  39. Linear Pair Postulate Two adjacent angles whose common sides are opposite rays. If two angles form a linear pair, then they are supplementary.

  40. Theorem 4.7 If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.

  41. Theorem 4.8 If two lines are perpendicular, then they intersect to form four right angles.

  42. Theorem 4.9 If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.

  43. Transversal A line that intersects two or more coplanar lines at different points.

  44. Theorem 4.10 Perpendicular Transversal Theorem If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.

  45. Theorem 4.11 Lines Perpendicular to a Transversal Theorem In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.

  46. Distance from a point to a line The length of the perpendicular segment from the point to the line. • Find the slope of the line • Use the negative reciprocal slope starting at • the given point until you hit the line • Use that intersecting point as your second • point. • Use the distance formula

  47. Congruent Figures All the parts of one figure are congruent to the corresponding parts of another figure. (Same size, same shape)

  48. Corresponding Parts The angles, sides, and vertices that are in the same location in congruent figures.

  49. Coordinate Proof Involves placing geometric figures in a coordinate plane.

  50. Side-Side-Side CongruencePostulate (SSS) If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.

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