(4 or more) Points that lie on the same plane.

Transcription

1 Geometry Chapter 1 Foundations Lesson 1: Understanding Points, Lines, and Planes Learning Targets LT1-1: Identify, name, draw and solve problems involving: points, lines, segments, rays and planes. Success Criteria Name points, lines & planes (with accurate notation). Draw segments and rays (with accurate notation). Identify points and lines in a plane (with accurate notation). Represent intersections (with accurate notation). What? Euclidean Geometry: Plane Geometry Coordinate Geometry Undefined Terms: point: So What? The math system attributed to Euclid of Alexandria, a Greek mathematician who write the Elements. Deals with objects that are flat and located somewhere in space. A system of geometry where the position of points are described using an ordered pair (x, y). These terms are defined to fit the branch of mathematics to be studied. Names a location; has no size; 0-D Collinear: line: plane: A straight path that has no thickness and extends forever in two directions. (set of points); 1-D A flat surface that has no thickness and extends forever in all directions; 2-D (3 or more) Points that lie on the same line. Coplanar: (4 or more) Points that lie on the same plane. Page 1

2 Segment: Endpoint: Ray: A part of a line consisting of two points (endpoints) and all points in between. A point at one end of a segment or the starting point of a ray. A part of a line that starts at one endpoint and extends forever in one direction. Opposite rays: Two rays that have a common endpoint and form a line. What? Postulate (axiom): So What? A statement accepted true without proof. An assumption. Point, Line, Plane Postulate: Unique Line Assumption Unique Plane Assumption Flat Plane Assumption Intersection of Lines and Planes Postulate: Line Intersection Plane Intersection 2 points determine a unique line. 3 points determine a unique plane. If two points lie in a plane, then the line containing the points also lies in the plane. 2 lines intersect at 1 point. 2 planes intersect at a line. Ex#1: Name points, lines & planes (with accurate notation). Using the figure at the right: A. Name three points that are collinear. B. Name three points that are coplanar. Page 2

3 Ex#2: Identify points and lines in a plane (with accurate notation). Use the figure at the right to name each of the following: A. a line containing point Q. B. a plane containing points P and Q. C. Name the line three different ways. Ex#4: Draw segments and rays (with accurate notation). A. Draw and label a segment with endpoints M and N. B. Draw and label opposite rays with common endpoint T. C. Is &TH the same as &HT? Explain. Ex#5: Represent intersections (with accurate notation). A. Sketch two lines intersecting in exactly one point. B. Sketch a line intersecting a plane. C. Sketch a line that is contained in Plane Q. D. Sketch three noncollinear points that are contained in Plane T. Page 3

4 Lesson 2: Measuring and Constructing Segments Learning Targets LT1-2: Calculate and construct midpoints, segment bisectors, and segment lengths. Success Criteria Find the length of a segment. Copy a segment. Use the Segment Addition Postulate. Apply measurements and constructions to realworld applications. Coordinate: What? So What? A point that corresponds to one (and only one) number on a ruler. Ruler Postulate: Distance: Congruent segments: The points on a line can be put into a one-to-one correspondence with the real numbers. The absolute value of the difference of the coordinates. AB means the distance from A to B Segments that have the same length. Segment Addition Postulate: If A is between M and P, then MA + AP = MP. Between: A point is between two others if its coordinate is greater than one endpoint and less than the other. Midpoint: (of a segment) is a point, M, on the line AB with AM = MB. Bisect: To cut into two equal parts. Page 4

5 Segment bisector: A ray, segment, or line that intersects a segment at its midpoint. Perpendicular bisector: A bisector of a segment that is also perpendicular to it. Construction: A way of creating a figure using only a compass and an unmarked straightedge. The length or measure of a segment always includes a unit of measure, such as inches, centimeter, etc. Ex#1: Find the length of a segment. Find the following.. A. KM= B. JN= C. IL= Caution: KM represents a number, while KM represents a geometric figure. Caution: Be sure to use equality for numbers (AB = YZ) and congruence for figures ( AB YZ ) 1.2 Construction: Congruent Segments You will need a clean sheet of paper and a compass for this construction. 1.2 Construction: Segment Bisector, Perpendicular Bisector, and Midpoint You will need a clean sheet of paper and a compass for this construction. Constructions: Page 5

6 Ex#2: Use the Segment Addition Postulate. A. If Y is between X and Z. If XY = 17 and XZ = 42, what is YZ? (Draw a diagram.) B. Find y and QP if P is between Q and R, QP = 2y, QR = 3y + 1, and PR = 21. (Draw a diagram.) C. K is the midpoint of JL. If JK = 3x 4 and LK = 5x 26, find x and JL. D. X is the midpoint of ZW. XW = 9 2a and ZW = 6a 9. Find ZX. Lesson 3: Measuring and Constructing Angles Learning Targets LT1-3: Name, measure, classify, and construct angles and their bisectors. Success Criteria Name angles using proper notation. Measure and classify angles. Use the Angle Addition Postulate. Find the measure of an angle. What? Angle Vertex Interior of an Exterior of an So What? Angle = a figure formed by the union of two rays. Vertex = the common endpoint T O P How to name an angle: Angles are named in various ways: You can name an angle by a single letter only when there is one angle shown at that vertex. When there is more than one angle at that vertex you must name the angle with three letters. Page 6

7 How to name an angle: Measure: The amount of openness of 2 rays that form an angle. Measured in degrees ( º ). Degree: ( º ) The common measure of an angle; 1/360 of a circle. Protractor Postulate: Given line AB and a point O on line AB, all rays that can be drawn can be put into a one-to-one correspondence with the real numbers 0 to 180. Congruent Angles: Angles that have the same measure. C G O T D A Angle Addition Postulate: If S is in the interior of PQR, then m PQS + SQR = PQR. Angle Bisector: A ray that divides an angle into two congruent angles. Page 7

8 Types of Angles: Acute Right Obtuse Straight M L N X Y Z P Q R A B C measure is 0 < m < 90 measure is 90 measure is 90 < m < 180 measure is 180 Ex#1: Measure and classify angles. Use the diagram to find the measure of each angle. Then classify each as acute, right, obtuse, or straight. A. DAB B. BAE C. EAD D. CAD 1.3 Construction: Congruent Angles 1.3 Construction: Angle Bisector Ex#2: Use the Angle Addition Postulate. A. If m LPR = 127, find each measure. Find m LPE Find m TPR B. Suppose m ATC = 145, m ATY = 6b + 10, and m CTY = 3b + 9. Find b. Find m ATY. Page 8

9 Ex#4: Use the Angle Addition Postulate. A. Suppose &IT bisects BIS. If m BIT= 37 Find m BIS. B. Suppose &IT bisects m BIS. m BIT = 12x + 3 and m TIS = 10x Find x and m BIS C. Suppose &IT bisects BIS. If m BIS = 44 and m TIS = 10x 13. Find x, Constructions: Page 9

10 Lesson 4: Pairs of Angles Learning Targets LT1-4: Classify pairs of angles as adjacent, vertical, complementary, or supplementary and solve problems involving them. Success Criteria Identify angle pairs and use them to solve problems. Find the measures of complements and supplements. Use complements and supplements to solve problems. Apply knowledge of angles and congruency to real-world applications. Identify vertical angles. Pairs of Angles Adjacent Angles: Two angles in the same plane with a common vertex and a common side, but no common interior points. Linear Pair: A pair of adjacent angles whose noncommon sides form opposite rays. Vertical Angles: Two nonadjacent angles formed by intersecting lines. Complementary Angles: Two angles whose measures have a sum of 90º. Supplementary Angles: Two angles whose measures have a sum of 180º. Linear Pair Theorem: If two angles form a linear pair, then they are supplementary. Page 10

11 Vertical Angle Theorem: If two angles are vertical angles, then they are equal in measure. Ex#1: Find the measures of complements and supplements. A. Tell whether the angles are only adjacent, adjacent B. Suppose two angles 3 and 4 are supplementary. and linear, or not adjacent. If m 3 = 47, what is m 4? 5 and 6 7 and SPU 7 and 8 Q R 8 6 P 7 5 S U T C. Suppose two angles 3 and 4 are complementary. If m 3 = x - 28, what is m 4? D. An angle is 10 more than 3 times the measure of its complement. Find the measure of the complement. Ex#2: Identify angle pairs and use them to solve problems. A. 1 and 2 form a linear pair. Suppose m 1 = 11n + 13 and m 2 = 5n 9. Find n and m 1. B. Suppose m 1 = 62, find as many angles as you can in the figure at the right. If m 1 = 10k, find as many angles as you can in the figure above. In geometry, figures are used to depict a situation. They are not drawn to reflect total accuracy of the situation. Page 11

12 From a figure, you can assume: 1. Collinearity and betweenness of points drawn on lines. 2. Intersection of lines at a given point. 3. Points in the interior of an angle, on an angle, or in the exterior of an angle. From a figure you cannot assume: 1. Collinearity of three of more points that are not drawn on lines. 2. Parallel lines. 3. Exact measures of angles and lengths of segments. 4. Measures of angles or lengths of segments are equal. Lesson 5: Using Formulas in Geometry Learning Targets LT1-5: Calculate basic perimeter and area of squares, rectangles, triangles, and circles. Success Criteria Find perimeter and area of figures. Apply geometric formulas to real-world applications. Find the circumference and area of a circle. Perimeter: What? So What? The sum of the side lengths of a figure. Area: Base and Height/ (Altitude) The number of nonoverlapping square units of a given size that exactly cover a figure. Base = any side of a triangle or polygon. Height/Altitude = a segment from a vertex that forms a right angle with a line containing the base. Diameter & Radius Diameter = A segment that passes through the center of a circle and whose endpoints are on the circle. Radius = a segment whose endpoints are the center of the circle and a points on the circle. d i a m e r a d i u Page 12

13 Circle The set of points in a plane at a certain distance (radius) from a given point (center). Circumference: Pi: The distance around a circle. C = 2πr or C = πd π = the ratio of a circles circumference to its distance across = C d 3.14 Perimeter and Area Formulas Rectangle Square Triangle P = P = P = A = A= A = Ex#1: Find perimeter and area of figures. A. B. 4 i n x x 6 i n 6 C. The Queens Quilt block includes 12 blue triangles. The base and height of each triangle are about 4 in. Find the approximate amount of fabric used to make the 12 triangles. D. The base of a rectangle is 5 more than 2 times its height. Find the perimeter and area of the rectangle. Page 13

14 Circumference and Area of a Circle C = A = r a d i u Ex#2: Finding the Circumference and Area of a Circle A. Find the exact area and circumference of a circle B. Find the area and circumference of a circle with a whose radius is 14 meters. diameter of 12cm. Round your answers to the nearest hundredth. Lesson 6: Midpoint and Distance in the Coordinate Plane Learning Targets LT1-6: Calculate distance and midpoint between two points in the coordinate plane. Success Criteria Find the coordinates of a midpoint. Find the coordinates of an endpoint. Use the distance formula. Find distances in the coordinate plane. What? Coordinate Plane/ Cartesian Plane: So What? A plane that is divided into 4 regions by a horizontal line (x-axis) and a vertical line (y-axis). Location of points is (x, y). Midpoint Formula: The midpoint M of AB with endpoints A 'x 1, y 1 ( and B 'x 2, y 2 ( is found by Distance Formula: The distance between two points ' x 1, y 1 ( and ' x 2, y 2 ( is Page 14

15 Pythagorean Theorem: In a right triangle, the sum of the squares of the lengths of the legs is equal to the squares of the length of the hypotenuse. a and b are called c is called Ex#1: Find the coordinates of a midpoint or endpoint. A. Find the coordinates of the midpoint of AB with B. M is the midpoint of XY. X has coordinates endpoints A(-8, 3) and B(-2, 7). (2, 7), and M has coordinates (6, 1). Find the coordinates of Y. Ex#2: Use the distance formula. Which segments are congruent? Show your work. You can also use the Pythagorean Theorem to find the distance between points in the coordinate plane. Page 15

16 Ex#3: Find distances in the coordinate plane. A. Graph the following points R(3, 4) and S(-2, -5) B. Use the distance formula and the Pythagorean Theorem to find the distance to the nearest hundredth. Distance Formula: Pythagorean Thrm: Ex#4: Find distances in the coordinate plane or using the distance formula. The four bases on a baseball field form a square with 90 foot sides. A player throws the ball from first base to a point located between third base and home plate and 10 feet from third base. What is the distance of the throw, to the nearest tenth? Lesson 7: Transformation in the Coordinate Plane Learning Targets LT1-7: Identify and graph reflections, rotations, and translations in the coordinate plane. Success Criteria Identify transformations from a picture and use arrow notation to describe it. Draw and identify transformations. Perform translations in the coordinate plane. Transformation: Preimage: Image: A change in the position, size, or shape of a figure. The original figure (or point). A The resulting figure (or point). Labeled with A'. Page 16

17 A transformation maps the preimage to the image. B B ' A ' A C ' C Transformations Reflection Rotation Translation A reflection is a transformation across a line, called the line of reflection. Each point and its image T A C T ' A ' are the same distance from the line of reflection. C ' A rotation is a transformation about a point P, called the center of rotation. Each point and its image are the same distance from P. D O D ' G G ' P O ' A translation is a transformation in which all the points of a figure move the same distance in the same direction. E S A L E ' S ' A ' L ' Notation: Notation: Notation: Ex#1: Name the transformation. Then use arrow notation to describe the transformation. A. B. M ' C. T I E A ' D ' T C C ' T ' T ' K E ' K ' M R E E ' R ' I ' A D Page 17

18 Ex#2: Drawing and Identifying the Transformations A. A figure has vertices E(2, 0), F(2, -1), G(5, -1) and H(5, 0). After a transformation, the image of the figure has vertices at E'(0, 2), F'(1, 2), G'(1, 5), and H'(0, 5). B. A figure has vertices at A(1, -1), B(2, 3), and C(4, -2). After a transformation the image of the figure has vertices at A'(-1, -1), B'(-2, 3), and C'(-4, -2). In the coordinate plane, to find the coordinates for the image, add a to the x-coordinates of the preimage and add b to the y-coordinate of the preimage. Translation Rule: ' x, y(%' x$a, y$b ( Ex#3: Perform translations in the coordinate plane. A. A figure has vertices A(-4, 2), B(-3, 4), and C(-1, 1). Find the coordinates for the image of ABC after the translation 'x, y(%'x$2, y 1( Draw the preimage and image. B. A figure has vertices J(1, 1), K(3, 1), M(1, -4), and L(3, -4). Find the coordinates for the image of JKLM after the translation 'x, y(%'x 2, y$4(. Draw the preimage and image. Page 18

19 Chapter 1 Homework Section Problems Tools 1.1 p #13-25, ruler 1.2 p #11, 12, 14, 15, 17, 18, 20-31, 34-39, 48 ruler 1.3 p #12-27, 29-32, 33, protractor 1.4 p #14-24, 27, 30, 34-38even, 40-42, p #10, 11, 13, 15, 17-25, 28, 29, 33, 35, 39, 41-43, p #12-18even, 21-25, 31, 32, 35, 36, 42, 43, 48 graph paper 1.7 p #8-17, 19-23, 25-27, 29-32, 41, 44 graph paper Page 19