NORTH HAVEN HIGH SCHOOL

NORTH HAVEN HIGH SCHOOL221 Elm StreetNorth Haven, CT 06473Geometry (Level 2 and Level 3)Summer Assignment 2017June 2017Dear Parents, Guardians, and Students,The Geometry curriculum builds on geometry concepts and vocabulary that were introduced in middle school and alsouses a number of techniques learned in Algebra I. Completing this packet will keep foundational geometry concepts andvocabulary fresh in students’ minds and provide a starting point for Geometry at the beginning of the year. Referencesheets are included in the packet to help complete the work.Please be sure that the completed packet is brought to school on the first day of class. The teacher will check the packetand students will receive a grade based on completion. Students must show work in this packet and complete all of theproblems to receive full credit. Calculators may be used when needed, but this cannot serve as a replacement forshowing work. If students have trouble with an item, they should skip it, and come back to it later, but persevere intrying to solve the problem.The mathematics department thanks you for your support and wishes you and your family a happy and restful summer!Sincerely,Ms. August, Mrs. Gaulin, and Mrs. OpramollaGeometry TeachersMrs. RombergMathematics Program Coordinatorromberg.tracey@north-haven.k12.ct.us

SAT Reference Information:The number of degrees of arc in a circle is 360.The number of radians of arc in a circle isThe sum of the measures in degrees of the angles of a triangle is 180.

Part 1 – Geometry Review/PreviewYou should know the following vocabulary from previous math classes. Please use the following referencesheets, to complete the problems on the subsequent geometry review/preview pages.The three undefined terms in geometry are: point, line, and plane. These are also called the “BuildingBlocks of Geometry” because everything is based on these three ideas. We are able to describe them butnot define them.Point – is a specific location in space with no size or shape.Line – is a set of points that go on indefinitely in both directions. Points on the same line are said to becollinear.𝑚symbolic notation Possible names:𝐵𝐴Q*named with any two points on the line ORan italicized single letter at the end of the line⃡𝐴𝐵⃡𝐵𝐴Line 𝑚Plane – is a flat surface with no edges and no boundaries. It has two dimensions. Objects on the sameplane are said to be coplanar.𝑅Q𝑄Possible names:Plane 𝑀Plane 𝑄𝑅𝑇Plane 𝑆𝑇𝑅𝑆𝑇𝑴*named by an upper case script letter ORany three non-collinear (not on the same line) points on the plane.Line segment – is part of a line containing two endpoints and all points between the endpoints.symbolic notation 𝐶𝐷*named with the two endpoints onlyPossible names:̅̅̅̅𝐶𝐷̅̅̅̅𝐷𝐶Congruent segments – two segments that have the same length. Congruent segments are marked indiagrams by tick marks or hashes.symbolic notation ̅̅̅̅𝑨𝑩 ̅̅̅̅𝑪𝑫 (said as “segment 𝑨𝑩 iscongruent to segment 𝑪𝑫)

Ray – is a portion of a line that extends from one point indefinitely in one direction.𝐸𝐻𝐹symbolic notation 𝑬𝑭𝐺symbolic notation 𝑮𝑯𝑅𝑄symbolic notation 𝑸𝑹*endpoint always named first then any other point on the ray.Opposite Rays – share the same end point and are collinear.𝐶𝐴𝐵𝑪𝑨 and 𝑪𝑩 are opposite raysAngle – two rays with the same endpoint. The common endpoint is called the vertex and the rays are𝐼called sides.𝐽symbolic notation 𝑱 is the vertex2𝐾Possible names: 𝐼𝐽𝐾 𝐾𝐽𝐼 𝐽 2*when naming angles the vertex is always the middle letter, the angle can also benamed by just the vertex letter or number inside the angle at the vertex.𝐵𝐶𝐴*You cannot name any of these angles 𝑨 because the vertex is sharedby three different angles.𝐷The most common unit of measure for angles is the degree. A protractor is used to measure angles. Use the90o angle as your reference angle when using a protractor.Congruent angles – two angles that have the same measure. Congruent angles are marked in diagramsusing arcs.40 40 symbolic notation 𝑮𝑰𝑩 𝑾𝑶𝑪 (said as “ Angle 𝑮𝑰𝑩 iscongruent to angle 𝑾𝑶𝑪”)

Angles can be classified by their degree measure.Right Angle:measures exactly 90 Acute Angle:Measures more than 0 and less than 90 Obtuse Angle:Measures more than 90 and less than 180 “Straight Angle”:Measures 180 Angle Pairs:Adjacent angles – two angles that share a common vertex and side, but have no common interiorPoints.* 𝑨𝑶𝑩 and 𝑩𝑶𝑪 are adjacent anglesComplementary angles – two angles whose measures have the sum of 90 Supplementary angles – two angles whose measures have the sum of 180

Linear Pair – two adjacent angles whose non-common sides are opposite rays (or form a straight angle)* 𝑫𝑨𝑪 and 𝑩𝑨𝑪 are a linear pair*linear pairs are supplementaryVertical angles – a pair of non-adjacent angles formed by the intersection of two lines* 𝑨𝑩𝑫 and 𝑬𝑩𝑪 are vertical angles*vertical angles are congruentAngle Bisector – a line, line segment, or ray which divides an angle into two equal parts.*𝑲𝑴 is an angle bisector of 𝑳𝑲𝑱Intersecting FiguresTwo lines intersect at one point only.Line and plane intersect at one point only.Two planes intersect at a line.Parallel lines – coplanar lines that never intersect. Parallel lines are marked in diagrams usingarrowheads.symbolic notation ⃡𝑱𝑲 ⃡𝑳𝑴 (said as “line 𝑱𝑲 is parallel toline 𝑳𝑴”)

Perpendicular lines – coplanar lines that intersect forming four right angles.Symbolic notation *𝒎 𝒍 (said as”line 𝒎 is perpendicular to line 𝒍)Skew lines – are not parallel, do not intersect and are not on the same plane.⃡ is skew to 𝑼𝑽⃡*𝑷𝑸Polygons- are closed figures, made up of line segments that meet only at their end points and are onone plane.# of SidesPolygon Name# of SidesPolygon nConvex polygon – a polygon that has all interior angles less than 180 . All the vertices point‘outwards’, away from the center.Concave polygon – a polygon that has one or more interior angle greater than 180 . Some verticespoint ‘inwards’, towards the center.

Triangles – There are special kinds of triangles. Triangles may be classified by their angle measures.Obtuse Triangle: hasone obtuse angle andtwo acute anglesRight Triangle: has oneright angle and twoacute anglesAcute Triangle: hasthree acute anglesEquilateral Triangle: specialkind of acute triangle, all 3angles measure 60 60 60 60 Triangles may also be classified by their side lengths.Scalene Triangle: no sides arethe same lengthIsosceles Triangle: at least twosides are the same lengthEquilateral Triangle: all threesides are the same length8 in7 in5 in5 cm3 in5 cm8 in5 in5 cmQuadrilaterals – There are special kinds of quadrilaterals.Trapezoid: has one pair of parallelsides (called bases shown to beparallel by use of arrows)Isosceles Trapezoid: has one pair ofparallel sides and the other twosides are the same length8 ft8 ftRectangle: parallelogram with fourright anglesParallelogram: has two pairs ofparallel sidesRhombus: parallelogram with foursides that are the same lengthSquare: parallelogram with fourright angles and four sides that arethe same length8 ft8 ft8 ft8 ftAll sidesmeasure5 feet

Part 2 – Geometric FiguresSketch and label each of the following geometric figures.1. ̅̅̅̅𝐺𝐸⃡2. 𝑂𝑀3. Plane 𝑇𝑅𝑌4. 𝐼𝑆5. right 𝑋𝐶𝐿⃡6. 𝑇𝐺7. Obtuse 𝐴𝑁𝐷8. Plane FUN9. Pentagon10. Acute 𝐷𝑀𝐻11. Scalene 𝐿𝐴𝑂12. Obtuse isosceles 𝑁𝐻𝑆13. Octagon14. Straight 𝐿𝑀𝑃15. Right 𝐽𝐷𝐺16. Hexagon17. Opposite rays 𝑀𝐽 and 18. Parallel lines ⃡𝐿𝐼 and⃡𝐶𝐴𝑀𝐿19. Perpendicular lines⃡𝐿𝐴 and 𝑅𝑄20. A pair of verticalangles, 𝐽𝑃𝑊 and 𝐾𝑃𝐶21. 𝑄𝑅𝑆 bisected by𝑅𝑇23. Collinear points 𝐴, 𝐵,𝐶, and 𝐷24. A linear pair, 𝐶𝐴𝐵and 𝐷𝐴𝐵22. A pair of adjacentangles, 𝐷𝑂𝐺 and 𝐿𝑂𝐷

You have been given one piece of information for each of the rows below. Complete the chart with theappropriate vocabulary term, definition, diagram or symbol.Vocabulary WordDescription / Definitiona.25.Diagramb.c.Line JKd.e.B26.f.Ag.h.i.k.l.A flat surface thatn.contains the non-colinear points SAG andextends infinitely inall directionso.p.r.Angle ABC whosemeasureis greater than 90 and less than 180 27.j.28.Ray PQm.29.q.Acute AngleJKL30.s.31.Symbolt.u.

Part 3 – AreaUse the SAT formula sheet to assist you in calculating the area of the following figures. Show formula(s) used andwork to support your answer. The first problem is complete and a few problems have been started for you.39.40.5 cm41.8 ft10 in9 ft12 cm9.3 inShape(s) :RectangleFormula: 𝐴𝑟𝑒𝑎 𝑏𝑎𝑠𝑒 ℎ𝑒𝑖𝑔ℎ𝑡Work:𝐴𝑟𝑒𝑎 12 𝑐𝑚. 5 𝑐𝑚.Shape(s) :Formula:Shape(s) :Triangle1Formula: 𝐴𝑟𝑒𝑎 𝑏 ℎ2Work:Answer:𝐴𝑟𝑒𝑎 60 𝑐𝑚2Answer:42.43.Work:Answer:44.5m22 cmShape(s) :Formula:Shape(s) :Formula:Work:Work:23.6 in14 inShape(s) :Formula:Work:Answer:45.Answer:10 cm46.Answer:9 cm47.16cmheight 13 cm8 cm38 cmShape(s) :TrapezoidFormula:1(𝑏𝑎𝑠𝑒1 𝑏𝑎𝑠𝑒2 )ℎ𝑒𝑖𝑔ℎ𝑡𝐴𝑟𝑒𝑎 2Work:Answer:15 cmShape(s) :Formula:Shape(s) :Formula:Work:Work:Answer:Answer:height 10 cm8 cm

48.49.Shape(s) :Shape(s) :Formula(s)& Work:Formula(s)& Work:Answer:Answer:A 3x10 rectangle has been removedfrom a 15x8 rectangle.Find the area of the shaded region.51. A 10’ x 9’ rectangle has been removed from a circle50.3 ft10 ftwith radius 6 ft. Find the area of the shaded region.6 ft8 ft10 ft15 ft9 ftShape(s) :Shape(s) :Formula(s)& Work:Formula(s)& Work:Answer:Answer:

52. Square with perimeter measuring 32 cm.Find the shaded area.53. A swimming pool measures 18’ x 32’. A cementwalkway that is 3’ wide is to be poured around thepool. How many square feet of cement will bepoured? Sketch a diagram and solve.Shape(s) :Formula(s)& Work:Answer:Answer:Part 4 – Radical ExpressionsYou must know (memorize!) the following facts regarding numbers that are perfect squares. We will focus ononly the positive values. When we have these values memorized, we can quickly calculate important values.12 122 432 942 1652 2562 3672 4982 6492 81102 100112 121122 144132 169142 196152 rethereforethereforethereforetherefore 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 Example #1: Simplifying radicals 72 36 2 6 2 6 2Example #2: Dividing Radicals 45 5 9 3Example #3: Squaring Radicals2(5 3) (5 3)(5 3) 25 9 25(3) 75Example #4: Rationalizing the denominator5 3 5 3 35 3 3 9 ( ) 5 33

Simplify the following expressions that contain perfect squares. Show all work in boxes to receive credit.Remember: Use your perfect squares to simplify.55. 19856. 4857. 9654. 5058. 16 3659. (3 2)260. 7261. 32 6Part 5 – Pythagorean TheoremUse the Pythagorean Theorem (𝑎2 𝑏 2 𝑐 2 ) to find the missing side of each right triangle. Give answer as awhole number or a simplified radical (no decimals).62.x 63.xx 25x91264.7x 65.x 10x845x

66.x 67.xx 3 2 53 6x68.Find the missing side and then the Perimeter and Area of the right triangle.x Perimeter 17xArea 1569.A 20 foot ladder is leaning against a house. The base of the ladder is 12 feet away from the house onthe ground. Draw a diagram and determine how far up the house the ladder will reach.70.Find the height and then the area of the isosceles triangle.1010Height ℎArea 12

Two lines intersect at one point only. Line and plane intersect at one point only. Two planes intersect at a line. Parallel lines – coplanar lines that never intersect. Parallel lines are marked in diagrams using arrowheads. symbolic notation ⃡ ⃡ (said as line is parallel to line )