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Geometry UNIT 1 Tools of Geometry Unit 1 Day 1 Points, Lines, and Planes Page 8: 13 – 31, 40, 43 - 48, 49, 52, 53 Unit 1 Day 2 Linear Measure Page 18: 10 – 19, 21, 23, 25, 27 - 32 Unit 1 Day 3 Distance and Midpoints Page 31: 13 – 31 left column, 33-53 left column Unit 1 Day 4 Angle measure Quiz Review Page 45: 1 - 15 Unit 1 Day 5 QUIZ Unit 1 Day 6 Angle Relationships Page 51: 22-24, 29-32, 36-47 Unit 1 Day 7 Two-Dimensional Figures Page 61: 12, 13, 15, 17, 18, 20, 27, 30abc, 40-42 Unit 1 Day 8 Three Dimensional Figures Page 71: 18 - 23, 51-55 Review TEST VOCABULARY Point Definition Congruent Segments Ray Interior Obtuse Complementary Convex Perimeter Edge Cone Line Defined Term Construction Opposite Rays Exterior Straight Supplementary n-agon Circumference Prism Sphere Plane Space Distance Angle Degree Adjacent Angles Perpendicular Equilateral Area Base Surface Area Collinear Line Segment Midpoint Side Right Angle Linear Pair Polygon Equiangular Polyhedron Pyramid Volume Intersection Betweenness Bisector Vertex Acute Vertical Angle Concave Regular Face Cylinder Points, Lines & Planes A point is a location. It has neither shape nor size. A line is made up of points and has no thickness or width. There is exactly one line through any two points. A plane is a flat surface made up of points that extends infinitely in all directions. There is exactly one plane through any three points not on the same line. Named by: a capital letter Named by: the capital letters representing two points on the line or a lower case script letter Named by: a capital script letter or by the letters naming three points that are not all on the same line. Example: Example: Example: ● Points, Lines & Planes On a subway map, the locations of stops are represented by points. The route the train can take is modeled by a series of connected paths that look like lines. The flat surface of the map on which these points and lines lie is representative of a plane. Unlike real-world objects that they model, shapes, points, lines & planes do not have any actual size. In geometry, point, line, & plane are considered undefined terms because they are only explained using examples and descriptions. Collinear points are points that lie on the same line. Noncollinear points do not lie on the same line. Coplanar points are points that lie in the same plane. Noncoplanar points do not lie in the same plane. Example: Name the geometric term modeled by each object. a) stripes on a sweater b) the corner of a box c) a sheet of notebook paper (sort of) 1. 2. 3. 4. 5. Recall From Algebra: Solving Quadratic Equations: Set the equation equal to zero Factor using GCF, DOPS, or Trinomial “Fish” for the answer Ex: q 2 – 2q + 1 = 0 Ex: 3x2 – 6x + 9 = 0 Ex: n2 + n = 30 Ex: x2 = 12x – 36 If K is between J and L: JK = x2 – 4x, KL = 3x – 2, JL = 28. Find JK and KL. Distance & Midpoints Distance Between Two Points The distance between two points is the length of the segment with those points as its endpoints. (REMEMBER: Length cannot be negative!) Distance Formula (on Number Line): The distance between two points is the absolute value of the difference between the coordinates. Example: See page 25 in your textbook (SMART BOARD). 1A) _________________ 1B) __________________ 1C) __________________ Example: Find the distance between (1,0) and (5, 0). Example: Find the distance between (5, 7) and (-3, 7). ______________ is another name for x-value; ______________ is another name for y-value Distance Formula (in the Coordinate plane) – compare to Pythagorean Theorem d= Example: ( x2 x1 ) 2 ( y2 y1 ) 2 or d = (x) 2 (y ) 2 Find the distance between J(4, 3) and K(-3, -7). Midpoint of a Segment The midpoint of a segment is the point halfway in between the endpoints of the segment. The coordinates of the midpoint can be found by finding the average of the abscissas and the average of the ordinates. Midpoint Formula x x 2 y1 y 2 M= 1 , 2 2 Example: Find the coordinates of the midpoint of a segment with the given coordinates. A(5, 12) and B(-4, 8) Example: Find the coordinates of the missing endpoint if P is the midpoint of EG . E(-8, 6) and P(-5, 10) Example: Find the measure of YZ if Y is the midpoint of XZ and XY = 2x – 3 and YZ = 27 – 4x. Angle Measure One of the skills needed for carpentry is how to cut a miter joint. This joint is created when two boards are cut at an angle to each other. One miscalculation in an angle measure can result in mitered edges that do not fit together. Measure and Classify Angles A ray is a part of a line. It has one endpoint and extends indefinitely in one direction. Rays are named by stating the endpoint first and then any other point on the ray. ▪ If you choose a point on a line, that point exactly two rays called opposite rays. Sine both rays share a common endpoint, opposite rays are collinear. ▪ An angle is formed by two noncollinear rays that have a common endpoint. The rays are called sides of the angle. The common endpoint is called the vertex. When naming angles using three letters, the vertex must be the middle of the three letters. You can name an angle by using a single letter only when there is exactly one angle at that vertex. Angles are measured in units called degrees. The degree results from dividing the distance around a circle into 360 parts. To measure an angle, use a protractor. Classify Angles Congruent angles have the same angle measure. A ray that divides an angle into two congruent angles is called an angle bisector. CONSTRUCT a congruent angle using the given angle and make a congruence statement. CONSTRUCT an angle bisector using the given angle and make a congruence statement. Angle Relationships Pairs of Angles Special Angle Pairs Adjacent angles are two angles that lie in the same plane and have a common vertex and a common side, but no common interior points. Example: Nonexample: A linear pair is a pair of adjacent angles with non common sides that are opposite rays. Example: Nonexample: Vertical angles are two nonadjacent angles formed by two intersecting lines. Example: Nonexample: Things that make you go “Hmmm…” Linear Pair vs. Supplementary Angles While the angles in a linear pair are always supplementary, some supplementary angles do not form a linear pair. Why? _____________________________________________________________________________ Angle Pair Relationships Vertical angles are congruent. Example: Complementary angles are two angles with measures that have a sum of 90º. Example: Supplementary angles are two angles with measures that have a sum of 180º. Example: The angles in a linear pair are supplementary. Example: Page 51 in textbook: #19 - 21 TWO DIMENSIONAL FIGURES VOCABULARY: Polygon: Vertex: Examples Convex Polygons “Non” Examples vs. Concave Polygons No points of lines are in the interior Some of the lines pass through the interior Equilateral Polygon: Number of Sides 3 4 5 6 7 8 9 10 11 12 n Equiangular Polygon: Regular Polygon: Polygon Formulas Triangle Square Rectangle Circle P= P= P= C= A= A= A= A= P = perimeter of the polygon b = base h = height A = area of figure l = length w = width C = circumference r = radius d = diameter 1. Name each polygon by the number of sides. Classify as concave or convex and regular or irregular. 2. Find the perimeter or circumference and the area of the figures. Round to the nearest tenth. 8m 17 m d = 12.8 cm 15 m 3. If I have vertices of A(-1, 2), B(3, 6) and C(3, -2), how might I find the perimeter and area of the triangle formed? THREE DIMENTIONAL FIGURES **Do page 80 #38 – 41 as GUIDED PRACTICE