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Reasoning in Geometry

Reasoning in Geometry. § 1.1 Patterns and Inductive Reasoning. § 1.2 Points, Lines, and Planes. § 1.3 Postulates. § 1.4 Conditional Statements and Their Converses. § 1.5 Tools of the Trade. § 1.6 A Plan for Problem Solving. 5 Minute-Check.

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Reasoning in Geometry

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  1. Reasoning in Geometry § 1.1 Patterns and Inductive Reasoning § 1.2 Points, Lines, and Planes § 1.3 Postulates § 1.4 Conditional Statements and Their Converses § 1.5 Tools of the Trade § 1.6 A Plan for Problem Solving

  2. 5 Minute-Check 1) Both answers can be calculated. Which one is right? What makes it right? What makes the other one incorrect? 2) Solve the equation. Check your answer. 3) If a dart is thrown at the circle to the right, what is the probability that it will land in a yellow sector? The odds?

  3. 5 Minute-Check Find the value or values of the variable that makes each equation true. g = 21 1. 2. x = 5 y = 4 or y = - 4 3. 4. z = - 2 5. 6 6. Find the next three terms of the sequence. 6, 12, 24, . . . 48, 96, 192

  4. Patterns and Inductive Reasoning What You'll Learn You will learn to identify patterns and use inductive reasoning. If you were to see dark, towering cloudsapproaching, you might want to take cover. Your past experience tells you that athunderstorm is likely to happen. When you make a conclusion based on a pattern of examples or past events, you are using inductive reasoning.

  5. X 2 X 2 X 2 + 1 X 2 + 3 + 5 + 7 + 9 X 2 Patterns and Inductive Reasoning You can use inductive reasoning to find the next terms in a sequence. Find the next three terms of the sequence: 24, 48, 96, 3, 6, 12, Find the next three terms of the sequence: 16, 23, 32 7, 8, 11,

  6. Patterns and Inductive Reasoning Draw the next figure in the pattern.

  7. d1 = 7.5 in. d2 = 7.5 in. Patterns and Inductive Reasoning conjecture A _________ is a conclusion that you reach based on inductive reasoning. In the following activity, you will make a conjecture about rectangles. 1) Draw several rectangles on your grid paper. 2) Draw the diagonals by connecting each corner with its opposite corner. Then measure the diagonals of each rectangle. 3) Record your data in a table Make a conjecture about the diagonals of a rectangle

  8. Patterns and Inductive Reasoning A conjecture is an educated guess. Sometimes it may be true, and other times it may be false. How do you know whether a conjecture is true or false? Try different examples to test the conjecture. If you find one example that does not follow the conjecture, then the conjecture is false. counterexample Such a false example is called a _____________. Conjecture: The sum of two numbers is always greater than either number. Is the conjecture TRUE or FALSE ? Counterexample: -5 + 3 = - 2 - 2 is not greater than 3.

  9. Patterns and Inductive Reasoning End of Lesson

  10. 5 Minute-Check Find the next three terms of each sequence. 1. 59, 63, 67 15.5, 20.5, 26.5 2. Draw the next figure in the pattern shown below. 3. 4. Find a counterexample for this statement: “The sum of two numbers is always greater than either addend.” -2 + 4 = 2 and 2 < 4 5) If a dart is thrown at the circle to the right, what is the probability that it will land in a shaded sector? The odds?

  11. Even the painting to the right is made entirely of small, carefully placed dots of color. Georges Seurat, Sunday Afternoon on the Island of LeGrande Jatte, 1884 - 1886 Points, Lines, and Planes What You'll Learn You will learn to identify and draw models of points, lines, andplanes, and determine their characteristics. Geometry is the study of points, lines, and planes and their relationships.Everything we see contains elements of geometry.

  12. A B Points, Lines, and Planes point A ____ is the basic unit of geometry. POINT: A point has no ____. size Points are named using capital letters. The points at the right are named point A and point B.

  13. A B The symbol for line AB is l Points, Lines, and Planes A ____is a series of points that extends without end in two directions. line infinite number LINE: A line is made up of an ______ _______ of points. arrows The ______ show that the line extends without end in both directions. A line can be named with a single lowercase script letter or by two points on the line. The line below is named line AB, line BA, or line l.

  14. m R T S or line m Points, Lines, and Planes 1) Name two points on line m. Possible answers: point R and point S point R and point T point S and point T 2) Give three names for the line. Possible answers: NOTE: Any two points on the line or the script letter can be used to name it.

  15. R T V S U 1) Name three points that are collinear. Possible answers: Points, Lines, and Planes collinear Three points may lie on the same line. These points are _______ . noncollinear Points that DO NOT lie on the same line are __________ . points R, S, and point T points U, S, and point V

  16. R T V S U 1) Name three points that are noncollinear. Possible answers: Points, Lines, and Planes collinear Three points may lie on the same line. These points are _______ . noncollinear Points that DO NOT lie on the same line are __________ . points R, T, and point U points R, S, and point V points R, S, and point U points R, T, and point V points R, V, and point U points S, T, and point V

  17. B A RAY: The starting point of a ray is called the ________. The symbol for ray AB is Points, Lines, and Planes Rays and line segments are parts of lines. ray A ___ has a definite starting point and extends without end in one direction. endpoint A ray is named using the endpoint first, then another point on the ray. The ray above is named ray AB.

  18. B A The symbol for segment AB is Points, Lines, and Planes Rays and line segments are parts of lines. line segment A ___________ has a definite beginning and end. LINE SEGMENT: A line segment is part of a line containing two endpoints and all points between them. A line segment is named using its endpoints. The line segment above is named segment AB or segment BA.

  19. C Points, Lines, and Planes 1) Name two segments. Possible Answers: D A B U 2) Name a ray. Possible Answers:

  20. Points, Lines, and Planes plane A _____ is a flat surface that extends without end in all directions. coplanar Points that lie in the same plane are ________. noncoplanar Points that do not lie in the same plane are ___________. PLANE: For any three noncollinear points, there is only one plane that contains all three points. A M A plane can be named with a single uppercase script letter or by three noncollinear points. B C The plane at the right is named plane ABC or plane M.

  21. C A B E D Hands On Place points A, B, C, D, & E on a piece of paper as shown. Fold the paper so that point A is on the crease. Open the paper slightly. The two sections of the paper represent different planes. Answers (may be others) A, B, & C 1) Name three points that are coplanar. ______________________ D, A, & B 2) Name three points that are noncoplanar. ______________________ A 3) Name a point that is in both planes. ______________________

  22. Points, Lines, and Planes End of Lesson

  23. r D F E C 5-Minute Check 1) Name three points on line r D, E, F 2) Give three other names for line r 3) Name two segments that have point F as an endpoint. 4) Name three different rays. 5) Are points C, E, and F collinear or noncollinear? noncollinear

  24. Postulates What You'll Learn You will learn to identify and use basic postulates about points, lines, and planes.

  25. C B A Q P l T m Postulates postulates Geometry is built on statements called _________. Postulates are statements in geometry that are accepted to be true. Postulate 1-1: Two points determine a unique ___. line There is only one line that containsPoints P and Q Postulate 1-2: If two distinct lines intersect, then their intersection is a ____. point Lines l and m intersect at point T Postulate 1-3: Three noncollinear points determine a unique _____. plane There is only one plane that contains points A, B, and C.

  26. 2) Name the intersection of Postulates Points A, B, and C are noncollinear. A 1) Name all of the different lines that can be drawn through these points. C B Point C

  27. A B D C Postulates 1) Name all of the planes that are represented in the figure. There is only one plane that contains threenoncollinear points. plane ABC (side) plane ACD (side) plane ABD (back side) plane BCD (bottom)

  28. D M E N Postulates Postulate 1-4: If two distinct planes intersect, then their intersection is a ___. line Plane M and plane N intersect in line DE.

  29. Name two planes that intersect in . F E A D H G B C Postulates Name the intersection of plane CDG and plane BCD. planes ADF and CDF

  30. Postulates End of Lesson

  31. A B D 4) Name two planes that intersect in . C 5) How many points do and have in common? 5-Minute Check 1) At which point or points do three planes intersect? At each of the points A, B, C, and D. 2) Name the intersection of plane ABC and plane ACD. 3) Are there two planes in the figure that do not intersect? No Planes ABD and BCD. One, (Point B)

  32. Conditional Statements and Their Converses What You'll Learn You will learn to write statements in if-then form and writethe converse of the statements. if-then statements In mathematics, you will come across many _______________. For Example: If a number is even, then it is divisible by two. If – then statements join two statements based on a condition: A number is divisible by two only if the number is even. conditional statements Therefore, if – then statements are also called __________ __________ .

  33. Conditional Statements and Their Converses Conditional statements have two parts. hypothesis The part following if is the _________ . conclusion The part following then is the _________ . If a number is even, then the number is divisible by two. a number is even the number is divisible by two. Hypothesis: Conclusion:

  34. Conditional Statements and Their Converses How do you determine whether a conditional statement is true or false? If it is the 4th of July(in the U.S.), then it is aholiday. True The statement is true becausethe conclusion follows fromthe hypothesis. If an animal lives in the water, then it is a fish. False You can show that the statement is false by givingone counterexample. Whales live in water, butwhales are mammals, not fish.

  35. Conditional Statements and Their Converses There are different ways to express a conditional statement.The following statements all have the same meaning. If you are a member of Congress, then you are a U.S. citizen. All members of Congress are U.S. citizens. You are a U.S. citizen if you are a member of Congress. You write two other forms of this statement: “If two lines are parallel, then they never intersect.” Possible answers: All parallel lines never intersect. Lines never intersect if they are parallel.

  36. Conditional Statements and Their Converses The ________ of a conditional statement is formed by exchanging the hypothesis and the conclusion. converse it has three angles Conditional: If a figure is a triangle, then it has three angles. a figure is a triangle Converse: If _______________, then ________________. NOTE: You often have to change the wording slightly so that the converse reads smoothly. Converse: If the figure has three angles, then it is a triangle.

  37. Conditional Statements and Their Converses Write the converse of the following statements.State whether the converse is TRUE or FALSE. If FALSE, give a counterexample: “If you are at least 16 years old, then you can get a driver’s license.” you can get a driver’s license you are at least 16 years old If ________________________, then _______________________. TRUE! “If today is Saturday, then there is no school. FALSE! there is no school today is Saturday If _______________, then ______________. We don’t have school on New Years day which may fall on a Monday.

  38. Conditional Statements and Their Converses End of Lesson

  39. 5-Minute Check If the power goes out, we will light candles. 1) Identify the hypothesis and conclusion of the statement. Hypothesis: the power goes out Conclusion: we will light candles 2) Write two other forms of the statement. 1) We will light candles if the power goes out. 2) Whenever the power goes out, we will light candles 3) Write the converse of the statement. If we light candles, then the power has gone out. 4) Is the converse you wrote for # 3 (above) true? NO! You could light candles for another reason, such as a birthday party.

  40. Tools of the Trade What You'll Learn You will learn to use geometry tools.

  41. Tools of the Trade As you study geometry, you will use some of the basic tools. straightedge A __________ is an object used to draw a straight line. A credit card, a piece of cardboard, or a ruler can serve as a straightedge. Determine whether thesides of the triangle arestraight. Place a straightedgealong each side of thetriangle.

  42. Tools of the Trade compass A _______ is another useful tool. A common use for a compass is drawing arcs and circles. (an arc is part of a circle)

  43. Use a compass to determine which segment is longer 2) Without changing the setting of the compass, place the point of the compass on B. The pencil point does not reach point D. Therefore, is longer. D B A C Tools of the Trade 1) Place the point of the compass on A and adjust the compass so that the pencil is on C. In geometry, you will draw figures using only a compass and a straightedge. These drawings are called ___________ . constructions

  44. Tools of the Trade Use a compass and straightedge to construct a six-sided figure. 1) Use the compass draw a circle. 2) Using the same compass setting, put the point on the circle and draw a small arc on the circle. 3) Move the compass point to the arc and draw another arc along the circle. Continue doing this until there are six arcs. 4) Use a straightedge to connect the points in order.

  45. Constructing the Midpoint You will learn to construct the midpoint of a line segment using only astraightedge and compass. 1) On your patty paper, draw two points. 2) Construct a line segment between the points 3) Fold the paper, and place one point on top of the other. This should produce a crease (fold mark) between the points. 4) Place the compass on one of the points and open it to over half way to the other point. 5) Repeat step 4 using the second point. 6) Connect the intersection of the two circles.

  46. Tools of the Trade End of Lesson

  47. A Plan for Problem Solving What You'll Learn You will learn to solve problems that involve the perimetersand areas of rectangles and parallelograms. distance around an object Perimeter is the _____________________. a line segment Perimeter is similar to ____________. number of square units needed to cover an object’s surface Area is the _______________________________________________. a plane Area is similar to ______.

  48. A Plan for Problem Solving In this section you will learn to solve problems that involve the perimetersand areas of rectangles and parallelograms. distance around a figure Perimeter is the ____________________. sum The perimeter is the ____ of the lengths of the sides of the figure. The perimeter of the room shown here is: 15 ft + 18 ft + 6 ft + 6 ft + 9 ft + 12 ft = 66 ft

  49. (of a rectangle) A Plan for Problem Solving Some figures have special characteristics. For example, the opposite sidesof a rectangle have the same length. This allows us to use a formula to find the perimeter of a rectangle.(A formula is an equation that shows how certain quantities are related.)

  50. 8 ft 17 ft (of a rectangle) A Plan for Problem Solving Find the perimeter of a rectangle with a length of 17 ft and a width of 8 ft. = 2(17 ft) + 2(8 ft) = 2(17 ft + 8 ft) = 34 ft + 16 ft = 2(25 ft) = 50 ft = 50 ft

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