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Section 1: Introduction to Geometry – Points, Lines, and PlanesTopic 1: Basics of Geometry - Part 1 .3Topic 2: Basics of Geometry – Part 2.5Topic 3: Midpoint and Distance in the Coordinate Plane – Part 1.7Topic 4: Midpoint and Distance in the Coordinate Plane – Part 2. 11Topic 5: Partitioning a Line Segment – Part 1 . 14Topic 6: Partitioning a Line Segment – Part 2 . 16Topic 7: Parallel and Perpendicular Lines – Part 1. 18Topic 8: Parallel and Perpendicular Lines – Part 2. 19Topic 9: Basic Constructions - Part 1 . 21Topic 10: Basic Constructions - Part 2 . 23Topic 11: Constructing Perpendicular Bisectors . 25Topic 12: Proving the Perpendicular Bisector Theorem Using Constructions . 27Visit MathNation.com or search "Math Nation" in your phone or tablet's app store to watch thevideos that go along with this workbook!Section 1: Introduction to Geometry — Points, Lines, and Planes1

The following Mathematics Florida Standards will be covered in this section:G-CO.1.1 - Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on theundefined notions of point, line, distance along a line, and distance around a circular arc.G-CO.3.9 - Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theoremsinclude: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruentand corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly thoseequidistant from the segment’s endpoints.G-CO.4.12 - Make formal geometric constructions with a variety of tools and methods. Copying a segment; copying anangle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector ofa line segment; and constructing a line parallel to a given line through a point not on the line.G-GPE.2.5 - Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.G-GPE.2.6 - Find the point on a directed line segment between two given points that partitions the segment in a givenratio.G-GPE.2.7 - Use coordinates to compute the perimeter of polygons and areas of triangles and rectangles.2Section 1: Introduction to Geometry — Points, Lines, and Planes

Section 1: Introduction to Geometry – Points, Lines and PlanesSection 1 – Topic 1Basics of Geometry – Part 1What is geometry?Points, lines, and planes are the building blocks of geometry.Draw a representation for each of the following and fill in theappropriate notation on the chart below.DescriptionGeometry means “ ,” and itinvolves the properties of points, lines, planes and figures.What concepts do you think belong in this branch ofmathematics?Why does geometry matter? When is geometry used in thereal A point is a preciselocation or place on aplane. It is usuallyrepresented by a dot.A line is a straight paththat continues in bothdirections forever.Lines are onedimensional.A plane is a flat,two-dimensionalobject. It has nothickness and extendsforever.DefinitionA line segment is aportion of a linelocated between twopoints.A ray is piece of a linethat starts at one pointand extends infinitelyin one direction.Section 1: Introduction to Geometry — Points, Lines, and Planes3

DefinitionRepresentationNotationA plane is a flat,two-dimensionalobject. It has nothickness and extendsforever.An angle is formed bytwo rays with thesame endpoint.The point where therays meet is called thevertex.Parallel lines are twolines on the sameplane that do notintersect.Perpendicular linesare two intersectinglines that form a 90 angle.What can you say about multiple points on a line segment?!"# %&'! !)* ,-./,0 %&%!20*304 4Segment Addition PostulateIf three points, 𝐴𝐴, 𝐵𝐵, and 𝐶𝐶, are collinear and𝐵𝐵 is between 𝐴𝐴 and 𝐶𝐶, then 𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵 𝐴𝐴𝐴𝐴.Section 1: Introduction to Geometry — Points, Lines, and Planes

Try It!2.Consider the descriptions of points, lines, and planes youhave learned so far.Use the word bank to complete the descriptions below.Draw a representation of each one.Word BankSection 1 – Topic 2Basics of Geometry – Part 2Let’s Practice!1.Consider the figure below.Parallel Planes Coplanar ParallelLines Collinear Non-collinearPoints that lie onDescription the same planeare .Points that lie onthe same line are.Select all the statements that apply to this figure.Drawingooooooo𝐴𝐴, 𝐵𝐵, 𝐶𝐶, and 𝐷𝐷 are coplanar in ℛ.𝐴𝐴, 𝐵𝐵, 𝐶𝐶, and 𝐹𝐹 are collinear.𝐴𝐴, 𝐵𝐵, and 𝑁𝑁 are collinear and coplanar in ℛ.𝐵𝐵 lies on 𝐴𝐴𝐴𝐴.𝐴𝐴, 𝐶𝐶 and 𝐹𝐹 are coplanar in ℛ.𝐶𝐶, 𝐷𝐷, 𝐸𝐸 and 𝐹𝐹 lie on ℛ.𝐴𝐴𝐴𝐴 𝑁𝑁𝑁𝑁 𝐴𝐴𝐴𝐴Section 1: Introduction to Geometry — Points, Lines, and Planes5

Try It!2.Plane 𝒬𝒬 contains 𝐴𝐴𝐴𝐴 and 𝐵𝐵𝐵𝐵, and it also intersects 𝑃𝑃𝑃𝑃 onlyat point 𝑀𝑀. Use the space below to sketch plane 𝒬𝒬.For points, lines, and planes, you need to know certainpostulates.A postulate is a statement that we take to beautomatically true. We do not need to provethat a postulate is true because it is somethingwe assume to be true.Let’s examine the following postulates A through F.A. Through any two points there is exactly one line.B.Through any three non-collinear points there is exactlyone plane.C. If two points lie in a plane, then the line containingthose points will also lie in the plane.D. If two lines intersect, they intersect in exactly onepoint.6E.If two planes intersect, they intersect in exactly oneline.F.Given a point on a plane, there is one and only oneline perpendicular to the plane through that point.Section 1: Introduction to Geometry — Points, Lines, and Planes

Section 1 – Topic 3Midpoint and Distance in the Coordinate Plane – Part 1Consider the line segment displayed below.𝐴𝐴10 cm𝐵𝐵The length of 𝐴𝐴𝐴𝐴 is centimeters.Øis an amount of space (in certainunits) between two points on a .Draw a point halfway between point 𝐴𝐴 and point 𝐵𝐵. Label thispoint 𝐶𝐶.What is the length of 𝐴𝐴𝐴𝐴?What is the length of 𝐶𝐶𝐶𝐶?Point 𝐶𝐶 is called the of 𝐴𝐴𝐴𝐴.Why do you think it’s called the midpoint?Section 1: Introduction to Geometry — Points, Lines, and Planes7

Let’s Practice!1.2.Consider the line segment below.Consider 𝑋𝑋𝑋𝑋 with midpoint 𝑅𝑅.𝑋𝑋a.𝐴𝐴𝑅𝑅𝑌𝑌a.What can be said of 𝑋𝑋𝑋𝑋 and 𝑅𝑅𝑅𝑅?(7𝑥𝑥 8) cm𝑀𝑀(9𝑥𝑥 8) cm𝐵𝐵If 𝐴𝐴𝐴𝐴 is 128 centimeters long, what is 𝑥𝑥?b. What is the length of 𝐴𝐴𝐴𝐴?b.8If 𝑋𝑋𝑋𝑋 is 2𝑥𝑥 5 inches long and 𝑅𝑅𝑅𝑅 is 22 inches long,what is the value of 𝑥𝑥?c.What is the length of 𝐵𝐵𝐵𝐵?d.Is point 𝑀𝑀 the midpoint of 𝐴𝐴𝐴𝐴? Justify your answer.Section 1: Introduction to Geometry — Points, Lines, and Planes

Try It!3.Consider the following graph.Diego and Anya live 72 miles apart. They both meet attheir favorite restaurant, which is 16𝑥𝑥 3 miles fromDiego’s house and 5𝑥𝑥 2 miles from Anya’s house.𝑦𝑦𝐵𝐵Diego argues that in a straight line distance, the restaurantis halfway between his house and Anya’s house. Is Diegoright? Justify your reasoning.𝑥𝑥𝐴𝐴Name the ordered pair that represents point 𝐴𝐴.Name the ordered pair that represents point 𝐵𝐵.Midpoint and distance can also be calculated on acoordinate plane.The coordinate plane is a plane that is divided intoregions (called quadrants) by a horizontal line ( )and a vertical line ( ).ØThe location, or coordinates, of a point are given byan ordered pair, .How can we find the midpoint of this line?The midpoint of 𝐴𝐴𝐴𝐴 is ( , ).Section 1: Introduction to Geometry — Points, Lines, and Planes9

Let’s consider points 𝑋𝑋 and 𝑌𝑌 on the coordinate plane below.𝑦𝑦Let’s Practice!4.Consider the line segment in the graph below.𝑦𝑦𝑌𝑌 (𝑥𝑥O , 𝑦𝑦O )𝑋𝑋 (𝑥𝑥N , 𝑦𝑦N )𝐵𝐵𝑥𝑥𝐴𝐴Write a formula that can be used to find the midpoint of anytwo given points.Find the midpoint of 𝐴𝐴𝐴𝐴.5.10𝑥𝑥𝑀𝑀 is the midpoint of 𝐶𝐶𝐶𝐶. 𝐶𝐶 has coordinates 1, 1 and𝑀𝑀 has coordinates 3, 5 . Find the coordinates of 𝐷𝐷.Section 1: Introduction to Geometry — Points, Lines, and Planes

Try It!6.𝑃𝑃 has coordinates 2, 4 . 𝑄𝑄 has coordinates ( 10, 12). Findthe midpoint of 𝑃𝑃𝑃𝑃.Section 1 – Topic 4Midpoint and Distance in the Coordinate Plane – Part 2Consider 𝐴𝐴𝐴𝐴 below.𝑦𝑦𝐴𝐴7.Café 103 is collinear with and equidistant from the MetricsSchool and the Angles Lab. The Metrics School is locatedat point 4, 6 on a coordinate plane, and Café 103 is atpoint 7, 2 . Find the coordinates of the Angles Lab.𝐵𝐵𝑥𝑥Draw point 𝐶𝐶 on the above graph at 2, 2 .What is the length of 𝐴𝐴𝐴𝐴?What is the length of 𝐵𝐵𝐵𝐵?Triangle 𝐴𝐴𝐴𝐴𝐴𝐴 is a right triangle. Use the Pythagorean Theoremto find the length of 𝐴𝐴𝐴𝐴.Section 1: Introduction to Geometry — Points, Lines, and Planes11

Let’s Practice!Let’s consider the figure below.1.𝑦𝑦𝑌𝑌 (𝑥𝑥O , 𝑦𝑦O )𝑋𝑋 (𝑥𝑥N , 𝑦𝑦N )𝑥𝑥Find the length of 𝐸𝐸𝐸𝐸.𝐸𝐸𝑥𝑥𝐹𝐹Write a formula to determine the distance of any linesegment.12𝑦𝑦Section 1: Introduction to Geometry — Points, Lines, and Planes

Try It!2.BEAT THE TEST!Consider triangle 𝐴𝐴𝐴𝐴𝐴𝐴 graphed on the coordinate plane.1.Consider the following �𝐴𝑥𝑥𝐶𝐶Find the perimeter of triangle ���𝐸𝑥𝑥𝐶𝐶Which of the following statements are true? Select all thatapply.oooooooR SThe midpoint of 𝐴𝐴𝐴𝐴 has coordinates ,O O.𝐷𝐷𝐷𝐷 is exactly 5 units long.𝐴𝐴𝐴𝐴 is exactly 3 units long.𝐹𝐹𝐹𝐹 is longer than 𝐸𝐸𝐸𝐸.The perimeter of quadrilateral 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is about 16.6 units.The perimeter of quadrilateral 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is about 18.8 units.The perimeter of triangle 𝐸𝐸𝐸𝐸𝐸𝐸 is 9 units.Section 1: Introduction to Geometry — Points, Lines, and Planes13

Section 1 – Topic 5Partitioning a Line Segment – Part 1Consider the following line segment where point 𝑃𝑃 partitionsthe segment into a 1: 4 ratio.What do you think it means to partition?𝐴𝐴𝑃𝑃14How many sections are between points 𝐴𝐴 and 𝑃𝑃?How can a line segment be partitioned?𝐵𝐵How many sections are between points 𝑃𝑃 and 𝐵𝐵?In the previous section, we worked with thepartitions a segment into a 1: 1 ratio., whichA ratio compares two numbers. A 1: 1 ratio isstated as, or can also be written as, “1 to 1”.How many sections are between points 𝐴𝐴 and 𝐵𝐵?In relation to 𝐴𝐴𝐴𝐴, how long is 𝐴𝐴𝐴𝐴?In relation to 𝐴𝐴𝐴𝐴, how long is 𝑃𝑃𝑃𝑃?Let’s call these ratios, 𝑘𝑘, a fraction that compares a part to awhole.Why does the midpoint partition a segment into a 1: 1 ratio?How can 𝐴𝐴𝐴𝐴 be divided into a 1: 3 ratio?𝐴𝐴14If partitioning a directed line segment into two segments,when would your ratio 𝑘𝑘 be the same for each segment?When would it differ?𝐵𝐵Section 1: Introduction to Geometry — Points, Lines, and Planes