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Reasoning in Geometry § 1.1 Patterns and Inductive Reasoning § 1.3 Postulates § 1.2 Points, Lines, and Planes § 1.4 Conditional Statements and Their Converses § 1.5 Tools of the Trade § 1.6 A Plan for Problem Solving 5 Minute-Check 1) Both answers can be calculated. Which one is right? What makes it right? What makes the other one incorrect? 2 3 X 6 30 2 3 X 6 20 2) Solve the equation. Check your answer. 4x 1 3 3 4x 2 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? favorable P total favorable O unfavorable 5 Minute-Check Find the value or values of the variable that makes each equation true. 1. 3g 63 2. 12 x 7 67 3. 2 y 2 32 4. 2 z 4 3z 6 0 5. If c 4 and d 3, what is the value of the expression 2 5d 3c ? g = 21 x=5 y=4 or y= -4 z=-2 6 6. Find the next three terms of the sequence. 6, 12, 24, . . . 48, 96, 192 Patterns and Inductive Reasoning You will learn to identify patterns and use inductive reasoning. If you were to see dark, towering clouds approaching, you might want to take cover. Your past experience tells you that a thunderstorm is likely to happen. When you make a conclusion based on a pattern of examples or past events, you are using inductive reasoning. 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: 3, 6, X2 24, 12, X2 X2 48, X2 96, X2 Find the next three terms of the sequence: 7, +1 8, 16, 11, +3 +5 23, +7 +9 32 Patterns and Inductive Reasoning Draw the next figure in the pattern. Patterns and Inductive Reasoning conjecture is a conclusion that you reach based on inductive reasoning. A _________ 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 Rectangle 1 Diagonal 1 Diagonal 2 7.5 inches 7.5 inches Make a conjecture about the diagonals of a rectangle d1 = 7.5 in. d2 = 7.5 in. 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. Patterns and Inductive Reasoning 5 Minute-Check Find the next three terms of each sequence. 1. 47, 51, 55 . . . 2. 5.5, 6.5, 8.5, 11.5, ... 15.5, 20.5, 26.5 3. Draw the next figure in the pattern shown below. 4. 59, 63, 67 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? Points, Lines, and Planes You will learn to identify and draw models of points, lines, and planes, and determine their characteristics. Geometry is the study of points, lines, and planes and their relationships. Everything we see contains elements of geometry. 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 point is the basic unit of geometry. A ____ POINT: A point has no ____. size A Points are named using capital letters. The points at the right are named point A and point B. B Points, Lines, and Planes A ____is line a series of points that extends without end in two directions. LINE: infinite _______ number of points. A line is made up of an ______ arrows show that the line extends without end in both The ______ directions. A line can be named with a single lowercase script letter or by two points on the line. The line below is named The symbol for line AB is l line AB, line BA, or line l. AB A B Points, Lines, and Planes m. 1) Name two points on line Possible answers: m R point R and point S T point R and point T point S and point T 2) Give three names for the line. Possible answers: RS RT ST or line S m NOTE: Any two points on the line or the script letter can be used to name it. 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 __________ R U 1) Name three points that are collinear. Possible answers: points R, S, and point T points U, S, and point V S V T 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 __________ R S U 1) Name three points that are noncollinear. Possible answers: points R, S, and point V points R, T, and point U points R, S, and point U points R, T, and point V points R, V, and point U points S, T, and point V V T Points, Lines, and Planes Rays and line segments are parts of lines. ray has a definite starting point and extends without end in one direction. A ___ A RAY: B endpoint The starting point of a ray is called the ________. A ray is named using the endpoint first, then another point on the ray. The ray above is named ray AB. The symbol for ray AB is AB Points, Lines, and Planes Rays and line segments are parts of lines. line segment has a definite beginning and end. A ___________ LINE SEGMENT: A line segment is part of a line containing two endpoints and all points between them. A B A line segment is named using its endpoints. The line segment above is named segment AB or segment BA. The symbol for segment AB is AB Points, Lines, and Planes 1) Name two segments. Possible Answers: AB, and AC , D A BD, and BC , B C 2) Name a ray. Possible Answers: AB, AC , DB, DU U Points, Lines, and Planes plane is a flat surface that extends without end in all directions. A _____ 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 A plane can be named with a single uppercase script letter or by three noncollinear points. The plane at the right is named plane ABC or plane M. M B C 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. D B A C E Answers (may be others) 1) Name three points that are coplanar. A, B, & C ______________________ 2) Name three points that are noncoplanar. D, A, & B ______________________ 3) Name a point that is in both planes. A ______________________ Points, Lines, and Planes 5-Minute Check 1) Name three points on line r r D D, E, F E C F 2) Give three other names for line r DE, DF , EF 3) Name two segments that have point F as an endpoint. 4) Name three different rays. DF , EF DC, DF (orDE), EF 5) Are points C, E, and F collinear or noncollinear? noncollinear Postulates You will learn to identify and use basic postulates about points, lines, and planes. Postulates postulates Geometry is built on statements called _________. Postulates are statements in geometry that are accepted to be true. Postulate 1-1: line Two points determine a unique ___. P Q There is only one line that contains Points P and Q Postulate 1-2: If two distinct lines intersect, point then their intersection is a ____. l T m Lines l and m intersect at point T Postulate 1-3: Three noncollinear points plane determine a unique _____. There is only one plane that contains points A, B, and C. A C B Postulates Points A, B, and C are noncollinear. A 1) Name all of the different lines that can be drawn through these points. AC CB BA 2) Name the intersection of AC, and CB Point C C B Postulates 1) Name all of the planes that are represented in the figure. There is only one plane that contains three noncollinear points. plane ABC (side) plane ACD (side) plane ABD (back side) plane BCD (bottom) A B D C Postulates Postulate 1-4: line If two distinct planes intersect, then their intersection is a ___. Plane M and plane N intersect in line DE. M D N E Postulates Name the intersection of plane CDG and plane BCD. DC Name two planes that intersect in DF. planes ADF and CDF F E A H B D G C Postulates 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. AC 3) Are there two planes in the figure that do not intersect? No 4) Name two planes that intersect in BD. Planes ABD and BCD. A 5) How many points do AB and BC have in common? One, (Point B) B D C Conditional Statements and Their Converses You will learn to write statements in if-then form and write the 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 __________ __________ . 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 even, then the number is divisible by two. Hypothesis: Conclusion: Conditional Statements and Their Converses How do you determine whether a conditional statement is true or false? Conditional Statement True or False Why? If it is the 4th of July (in the U.S.), then it is a holiday. True The statement is true because the conclusion follows from the hypothesis. If an animal lives in the water, then it is a fish. False You can show that the statement is false by giving one counterexample. Whales live in water, but whales are mammals, not fish. 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. Conditional Statements and Their Converses The ________ converse of a conditional statement is formed by exchanging the hypothesis and the conclusion. angles Conditional: If a figure is a triangle, triangle then it has three angles. 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. 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 then _______________________. you are at least 16 years old If ________________________, TRUE! “If today is Saturday, then there is no school. FALSE! there is no school then ______________. today is Saturday If _______________, We don’t have school on New Years day which may fall on a Monday. Conditional Statements and Their Converses 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. Tools of the Trade You will learn to use geometry tools. Tools of the Trade As you study geometry, you will use some of the basic tools. straightedge is an object used to draw a straight line. A __________ A credit card, a piece of cardboard, or a ruler can serve as a straightedge. Determine whether the sides of the triangle are straight. Place a straightedge along each side of the triangle. Tools of the Trade A compass _______ is another useful tool. A common use for a compass is drawing arcs and circles. (an arc is part of a circle) Tools of the Trade Use a compass to determine which segment is longer AC or BD 1) Place the point of the compass on A and adjust the compass so that the pencil is on C. 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, BD is longer. D In geometry, you will draw figures using only a compass and a straightedge. These drawings are called ___________ constructions . A B C Tools of the Trade Use a compass and straightedge to construct a six-sided figure. 1) Move 2) 3) 4) Use a Using the the the straightedge compass compass same compass draw point to connect a to circle. setting, thethe arcput points and the draw in point order. another on the arc circle along andthe draw circle. a small arc on Continue doing thethis circle. until there are six arcs. Constructing the Midpoint You will learn to construct the midpoint of a line segment using only a straightedge 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. Tools of the Trade A Plan for Problem Solving You will learn to solve problems that involve the perimeters and 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 ______. A Plan for Problem Solving In this section you will learn to solve problems that involve the perimeters and areas of rectangles and parallelograms. distance around a figure Perimeter is the ____________________. sum of the lengths of the sides of the figure. The perimeter is the ____ The perimeter of the room shown here is: 15 ft + 18 ft + 6 ft + 6 ft + 9 ft + 12 ft = 66 ft A Plan for Problem Solving Some figures have special characteristics. For example, the opposite sides of 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.) Perimeter 2l 2w (of a rectangle) 2(l w) A Plan for Problem Solving Find the perimeter of a rectangle with a length of 17 ft and a width of 8 ft. 8 ft 17 ft Perimeter 2l 2w or 2(l w) (of a rectangle) = 2(17 ft) + 2(8 ft) = 2(17 ft + 8 ft) = 34 ft + 16 ft = 2(25 ft) = 50 ft = 50 ft A Plan for Problem Solving Another important measure is area. the number of square units needed to cover its surface The area of a figure is ____________________________________________. The area of the rectangle below can be found by dividing it into 18 unit squares. 3 6 The area of a rectangle can also be found by multiplying the length and the width. A Plan for Problem Solving The area “A” of a rectangle is the product of the length l and the width w. A lw w l Find the area of the rectangle A lw A (14in)(10in) A 140in 2 10 in. 14 in. The area of the rectangle is 140 square inches. NOTE: units indicate area is being calculated (in)(in) in 2 Plan for Problem Solving Because the opposite sides of a parallelogram have the same length, rectangle the area of a parallelogram is closely related to the area of a ________. height base height The area of a parallelogram is found by multiplying the base ____ and the ______. Base – the bottom of a geometric figure. Height – measured from top to bottom, perpendicular to the base. A Plan for Problem Solving Find the area of the parallelogram: A bh 51 (4m) m 10 204 2 m 10 2 2 20 m 5 4.3 m 4m 5 1 m 10 §1.6 A Plan for Problem Solving §1.5 Tools of the Trade