NYS COMMON CORE MATHEMATICS CURRICULUM Unit 1

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMGEOMETRYCongruence, Proof and ConstructionsNAMEDATE9/79/89/119/12Lesson #1233Page(s)2-34-67-9TopicVocab and postulatesConstruct Equilateral TrianglesCopy Angles and Construct Angle BisectorsFinish Lesson 3QUIZPractice AssignmentProblem Set Unit 1 Lesson 1Problem Set Unit 1 Lesson 2Problem Set Unit 1 Lesson 3None9/139/149/1544510-11NoneFinish Class workProblem Set Unit 1 Lesson 59/189/199/209/21614-157816-1819-21Construct Perpendicular BisectorsFinish lesson 4 and In Class assignmentConstructions of Parallel and PerpendicularlinesPoints of ConcurrencyQUIZAngles and Lines at a 5-26Auxiliary LinesSum of Angles in A TriangleAngles in TrianglesReviewTESTProblem Set Unit 1 Lesson 9Problem Set Unit 1 Lesson 10Problem Set Unit 1 Lesson 11TICKET IN- Study!!!!None12-131Problem Set Unit 1 Lesson 6NoneProblem Set Unit 1 Lesson 7Problem Set Unit 1 Lesson 8

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMLesson 1: Vocab and Postulates with Planes and LinesTerm and definitionLink to exploredefinitionPoint- A precise location or place on aplane. Usually represented by a dot.http://www.mathopenref.com/point.htmlLine- A geometrical object that isstraight, infinitely long and lSegment- A straight line which linkstwo points without extending htmlRay- A portion of a line which starts ata point and goes off in a particulardirection to ear- Points that lie on the samestraight e- A flat surface that is infinitelylarge and with zero lanar- Objects are coplanar if theyall lie in the same le- The set of points that are allequidistant from a fixed point, thecenter - A line from the center of acircle to a point on the am2

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Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMLesson 2: Construct an Equilateral Triangle IIWatch the short animation on constructing an equilateral uct-equitriangle.htmlBased on the animation, list the precise steps used to create any equilateral triangle:TRY IT! Construct an equilateral triangle using AB as one side of the triangle.4

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMExploratory Challenge 1You will need a compass and a straightedgea.) Using the skills you have practiced, construct three equilateral triangles, where the first and second triangles share a commonside, and the second and third triangles share a common side. Clearly and precisely list the steps needed to accomplish thisconstruction.CONSTRUCTION:b.) Continuing with the process same process, construct a regular hexagon (think about how many equilateraltriangles you would need to join together)c.) Draw a circle such that the hexagon is inscribed inside the circle.Inscribe:d.) Connect three points on the hexagon to form an equilateral triangle. Explain how we would know that thetriangle would have to be equilateral:e.) What other shapes could you make by joining equilateral triangles?5

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMExploratory Challenge 2:Construct two isosceles triangles one acute isosceles triangle and one obtuse isosceles triangle where AB is the base ofthe triangle.Define: Isosceles triangle-Acute triangle-Obtuse triangle-6

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMLesson 3: Angle Vocab, Copy and Bisect an AngleTerm and definitionLink toexploredefinitionDiagramAngle- The union of two rays sharingthe same ee- 1/360th of a e Angle- An angle whose measureis greater than 0o and less than se Angle- An angle whosemeasure is greater than 90o and lessthan lex Angle- An angle whosemeasure is greater than acent Angles- Two angles thatshare a common side and cent.htmlComplementary Angles- Two anglesthat add to 90o. Often timescomplementary angles are adjacent inwhich case they form a right y.htmlSupplementary Angles- Two anglesthat add to 180o. When the two anglesare adjacent they are called html7

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMAngle Bisector- A ray that divides anangle into two equal mlMidpoint- A point on a segment thatis equidistant from each egment Bisector-A line, ray orsegment which cuts another linesegment into two equal Video: Watch the video Angles and TrimWhile watching the video make note of how to bisect an angle.Experiment with the angles below to determine the correct steps for the construction.List the steps you used to bisect the angles:8

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMCritical thinking:-Constructing an is fundamentally the same as constructing a whole circle.-The angle bisector could also be called the line of since the same procedure was done toboth sides of the angle.Example 2: Investigate How to Copy an AngleYou will need a compass and a straightedge.Together with a partner, copy the angle onto the blank space below by following the steps given:9

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMLesson 4: Construct a Perpendicular BisectorTerm and definitionLink toexploredefinitionDiagramEquidistant-A point P is equidistantfrom others if it is the same distancefrom rpendicular-A line is perpendicularto another if it meets or crosses it atright angles (90 endicular Bisector-A line whichcuts a line segment into two equalparts at 90 htmlConstruct a perpendicular bisector of a line segment using a compass and straightedge. Using what you know about theconstruction of an angle bisector,ABPrecisely describe the steps you took to bisect the segment.10

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMDiscussion:Watch the animation of a perpendicular bisector here isect.html). Are the steps taken in the animation the same or different than the steps above?Now that you are familiar with the construction of a perpendicular bisector, we must make one last observation. Inthe diagram CE is the perpendicular bisector of AB. Using your compass, examine the following pairs of segments:I.̅̅̅̅𝐴𝐶 , ̅̅̅̅𝐵𝐶II.̅̅̅̅𝐴𝐷, ̅̅̅̅𝐵𝐷III.̅̅̅̅𝐴𝐸 , ̅̅̅̅𝐵𝐸CDBased on your findings, fill in the observation below.BAEObservation:Any point on the perpendicular bisector of a line segment isfrom the endpoints of the line segment.What kind of triangles are triangles ACB, ADB, and AEB?11

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMLesson 5: Constructions of Perpendicular and Parallel linesMathematical Modeling ExerciseYou know how to construct the perpendicular bisector of a segment. Now you will investigate how to construct a perpendicularto a line ℓ from a point 𝐴 not on ℓ. Think about how you have used circles in constructions so far and why the perpendicularbisector construction works the way it does. Watch the animation otline.html) to see the construction, complete the construction on your own and write the steps.AℓSteps:12

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMDiscussion:What other shapes can we construct now that we know how to make parallel and perpendicular lines?13

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMLesson 6: Points of ConcurrencyDiscussion1.) When three or more lines intersect in a single point, they are , and the point of intersection is thepoint of .2.) The point of concurrency of the three perpendicular bisectors is the of thetriangle.EXPLORE: We will use ml to explore what happens when the triangle isright or obtuse. Sketch the location of the circumcenter on the triangles below:3.) The circumcenter of 𝐴𝐵𝐶 is shown below as point 𝑃.Mark all of the right angles in the diagram.Mark the congruent segments in the diagram.14

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMEXAMPLE 2Use the triangle below to construct the angle bisectors of each angle in the triangle.ABCThe construction of the three angle bisectors of a triangle also results in a point of concurrency, which we call the.EXPLORE: We will use http://www.mathopenref.com/triangleincenter.html to explore what happens when the triangle is right orobtuse. Sketch the location of the incenter on the triangles below:The triangle below shows an incenter. Using the incenter, draw a circle inscribed inside the triangle.15

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMLesson 7: Solve for Unknown Angles—Angles and Lines at a PointOpening Exercise- Determine the measure of the missing angle in each diagram.Types of AnglesLinkWrite an equation using the angles:Vertical Angles- A pair of non-adjacentangles formed by the intersection oftwo straight lAngles Forming a Right angle-Angles Forming a Straight line-Angles around a Point-16

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMExample 1Find the measures of each labeled angle. Give a reason for your solution.AngleAnglemeasureReason a b c d e17

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMExercisesIn the figures below, ̅̅̅̅𝐴𝐵 , ̅̅̅̅𝐶𝐷 , and ̅̅̅̅𝐸𝐹 are straight line segments. Find the measure of each marked angle or find the unknownnumbers labeled by the variables in the diagrams. Give reasons for your calculations. Show all the steps to your solution.1.3.2.4.5.6.18

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMLesson 8: Solve for Unknown Angles—TransversalsGiven a pair of lines ⃡𝐴𝐵 and ⃡𝐶𝐷 in a plane (see the diagram below), a third line 𝐸𝐹 is called a transversal if it intersects 𝐴𝐵 at asingle point and intersects ⃡𝐶𝐷 at a single but different point.Follow the link to explore the definitions and fill in the htmlPairs of AnglesIdentify a pair of angles in the above diagram for each term and writean equation using the pair of anglesCorresponding Angles-Alternate Interior Angles-Alternate Exterior Angles-Same Side Interior Angles (See Interior Angles of aTransversal)EXAMPLE 119

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMEXAMPLE 220

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Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMLESSON 9: AUXILARY LINES and angles in trianglesSometimes adding a line to a picture can be helpful.In this case we call the line an .In this figure, we use an auxiliary line to find the measures of 𝑒 and 𝑓(how?), then add the two measures together to find the measure of 𝑊.What is the measure of 𝑊?Note: An auxiliary line can be drawn anywhere in the diagram needed but is usually most helpful when drawn parallelto and between the two given parallel lines.ExercisesIn each exercise below, find the unknown (labeled) angles. Give reasons for your solutions.1.2.3.4.5.6.22

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUM7.8.23

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMLesson 10: Sum of Angles in a TriangleExercisesIn each figure, determine the measures of the unknown (labeled) angles. Give reasons for your calculations.1.2.3.4.5.6.24

Unit 1 Module 1NYS COMMON CORE MATHEMATICS CURRICULUMLesson 11: Angles in TrianglesExercises:25

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Problem Set Unit 1 Lesson 5 9/18 6 14-15 Points of Concurrency Problem Set Unit 1 Lesson 6 9/19 QUIZ None 9/20 7 16-18 Angles and Lines at a point Problem Set Unit 1 Lesson 7 9/21 8 19-21 Transversals Problem Set Unit 1 Lesson 8 9/22 9 22-23 Auxiliary Lines Problem Set Unit 1 Lesson 9