Unit 8 Plane Geometry

Transcription

1 Unit 8 Plane Geometry Grade 9 pplied Lesson Outline *Note: This unit could stand alone and be placed anywhere in the course. IG PITURE Students will: investigate properties of geometric objects using dynamic geometry software and manipulatives; illustrate and explain the relationship between angles formed by parallel lines cut by a transversal and interior and exterior angles of triangles and quadrilaterals; determine some properties of sides and diagonals of quadrilaterals. Note: Students may have a very broad range of experience with using The Geometer s Sketchpad 4. Skills can be taught as they are needed for each lesson, or alternatively, Introduction to The Geometer s Sketchpad 4 file (included in Day 2) could be used at the beginning of the unit. Math Learning Goals Describe the properties and relationships of the angles formed by parallel lines cut by a transversal. Review angles, triangles, and parallel lines through exploration. uild skills required for future use of The Geometer s Sketchpad 4 (GSP 4). Explore geometrical concepts (angles, triangles, parallel lines). uild skills required for future use of GSP 4. uild investigation skills by exploring geometric concepts, using GSP 4. Develop communication skills and geometric vocabulary. Investigate the sum of the interior and exterior angles of triangles and quadrilaterals using GSP 4 and demonstration. Develop skills with GSP 4 in preparation for summative assessments. Practise solving problems using the geometry explored in previous lessons. Make connections to solving equations. Expectations LR1.03 LR2.02 LR4.04 MG3.01 MG3.02 MG3.03 N2.07 Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 1

2 8.1: Plane Geometry The asics ngle Sum of a Triangle Theorem (STT) Description Diagram Example The three interior angles of a triangle add up to 180 a Find the missing angle. x b c a + b + c = x = x = 85 (STT) Find the missing angles using the angle sum of a triangle theorem y 25 x z Isosceles Triangle Theorem (ITT) Description Diagram Example Isosceles triangles have two sides equal and two angles equal. (The two equal angles occur where the equal sides meet the third side.) Find the missing angles or sides using the isosceles triangle theorem. x x 6 cm y 20 x x = 20 (ITT) y = 6 cm (ITT) cm y x 80 z Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 2

3 omplementary ngle Theorem (T) Description Diagram Example omplementary angles are two or more joined angles that add up to 90 Find the missing angles using the complementary angle theorem x y x + y = 90 Find the missing angle. 60 x x = = 30 (T) x y 33 z Supplementary ngle Theorem (ST) Description Diagram Example Supplementary angles are two or more joined angles that add up to 180. x y x + y = 180 x x = x = 40 (ST) 140 Find the missing angles using the supplementary angle theorem x y 147 z Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 3

4 Opposite ngle Theorem (OT) Description Diagram Example Opposite angles are equal angles that are across from each other when two lines intersect. x y y x y x x = 39 (OT) y = 141 (OT) Find the missing angles using the opposite angle theorem x y z Exterior ngle Theorem (ET) Description Diagram Example The exterior angles in a triangle are equal to the sum of the two opposite (remote) interior angles in the triangle. Find the missing angles using the exterior angles theorem. b a a + b Find the missing angle x x = x = 145 (ET) x 70 y z Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 4

5 Parallel Line Properties ORRESPONDING NGLES (F) Description Diagram Example When 2 parallel lines are intersected by a third line (transversal), an F pattern can be found. This pattern shows corresponding angles. x x Find the missing angle using corresponding angles. y 70 y y 110 x x = 70 (F) y = 110 (F) Find the missing angles using corresponding angles (F-pattern) on the diagram. Draw the F-pattern on the diagram z y 65 x Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 5

6 Parallel Line Properties - LTERNTE NGLES (Z) Description Diagram Example When 2 parallel lines are intersected by a third line (transversal), a Z pattern can be found. This pattern shows alternate angles. x x Find the missing angle using alternate angles. 74 x y y x = 74 (Z) Find the missing angles using alternate angles (Z pattern) on the diagram. Draw the Z-pattern on the diagram. z y 60 x Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 6

7 Parallel Line Properties O-INTERIOR NGLES () Description Diagram Example When 2 parallel lines are intersected by a third line (transversal), a pattern can be found. This pattern shows co-interior angles. o-interior angles add up to 180. x + y = 180 x y Find the missing angle using co-interior angles. 80 x x y x = 180 o 80 o x = 100 () Find the missing angles using co-interior angles (-pattern) on the diagram. Draw the -pattern on the diagram. 120 y 120 z 60 x Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 7

8 8.2: Theorems Practice Sheet Define each principle and determine the unknown angles. 1. Supplementary ngles - x 85 x = = 2. omplementary ngles - m 30 m = = 3. Opposite ngle Theorem - r 70 r = = 4. The Interior ngles of a Triangle - q q = = 5. Isosceles Triangle Theorem - 75 a a = = Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 8

9 6. Equilateral Triangles - x x = = 7. Exterior ngle Theorem b b = = 8. Parallel Lines a) orresponding ngles - 68 m m = b) lternate ngles - 83 w w = c) o-interior ngles - x 72 x = = Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 9

10 8.3: Plane Geometry Worksheet 1. lassify each triangle and find the measurement of all unknown angles. Include reasons for each angle. 2. Find the measures of the unknown angles. Include reasons for each angle. Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 10

11 8.4: Plane Geometry Record Sheet Use this page to record your observations and conclusions from the Plane Geometry GSP 4 file. Determine the unknown angle in the right column. Give reasons for your answer. Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 11

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14 8.5: Word Wall List acute angle acute triangle adjacent angles alternate angles bisector circle co-interior angles complementary angles congruent corresponding angles diagonal diameter equilateral triangle exterior angle hexagon interior angle isosceles triangle kite line line segment midpoint obtuse angle obtuse triangle octagon opposite angle parallelogram pentagon perpendicular perpendicular bisector point polygon quadrilateral radius ray rectangle reflex angle rhombus right angle right angle triangle scalene triangle side similar figures supplementary angles transversal Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 14

15 Plane Geometry (GSP 4 file) Plane Geometry.gsp Plane Geometry ngles Explore each of the ten activities. (Start by clicking on number 1.) Supplementary ngles omplementary ngles Opposite ngles 1) Supplementary ngles Given: Explore: is a straight angle. Measure D and D. D Main Menu Each activity has a button which will bring you back to this page. If you need help on how to use Geometer's Sketchpad go to the "How do I...?"page by pressing the button: How do I...? Triangles Sum of the Interior ngles in a Triangle Isosceles Triangles Equilateral Triangles Exterior ngle of a Triangle Parallel Lines orresponding ngles (F-Pattern) lternate ngles (Z-Pattern) o-interior ngles (-Pattern) onclude: How do I...? alculate the sum of these two angles. Drag point D. What is the sum of the two angles? Main Menu Next page 2) omplementary ngles 3) Opposite ngle Theorem Given: Explore: onclude: O is a right angle. Measure O and O. How do I...? alculate the sum of these two angles. nimate the diagram by pressing the button below. You can pause the animation by pressing the button again. nimate What is the sum of the two angles? O Given: intersects D at O. Explore: Measure O and DO. How do I...? nimate the diagram by pressing the button below. You can pause the animation by pressing the button again. nimate onclusion: What do you notice about O and DO? O D Main Menu Next page Main Menu Next page 4) The Sum of the Interior ngles of a Triangle Given: with the interior angles identified. Explore: lick on the action button below. Show the Sum of the ngles Reset b a c 5) Isosceles Triangle Theorem Given: Isosceles, where = Explore: Measure the angles opposite the equal sides ( and ) How do I...? Drag each of the vertices. Drag each vertex of the triangle. onclude: What do you notice about the angles opposite the equal sides? (Make sure the coloured sections marking interior angles are always inside the triangle.) onclude: What is the sum of the interior angles of a triangle? Main Menu Next page Main Menu Next page Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 15

16 Plane Geometry (GSP 4 file) (continued) 6) Equilateral Triangles 7) Exterior ngle Theorem Given:, where = = Explore: Measure each of the interior angles then drag point. Given: and exterior angle D. Explore: Measure the non-adjacent angles &. How do I...? onclude: What do you notice about each of the interior angles? How do I...? alculate the sum of these two angles. Measure the exterior angle D. Drag point. onclude: What do you notice about the sum of the two non-adjacent angles, &, inside the triangle and the exterior angle, D? D Main Menu Next page Main Menu Next Page 8) Parallel Line Theorem - orresponding ngles Reset Given: Explore: is parallel to D with transversal PQ. Measure the corresponding angles WX and DXP. How do I...? Drag points and X. onclude: What do you notice about the corresponding angles WX and DXP? More... Explain why this relationship is called the F-pattern. Q W X P D 9) Parallel Line Theorem - lternate ngles Reset Z Pattern Given: is parallel to D with transversal PQ. Explore: Measure the alternate angles WX and WX. How do I...? Drag points and X. onclude: What do you notice about the alternate angles WX and WX? More... Explain why this relationship is called the Z-pattern. There are two "Z-patterns" in this diagram. an you find them? Help Q W X P D There are four "F-patterns" in this diagram. an you find them? Help Main Menu Next page Main Menu Next page 10) Parallel Line Theorem - o-interior ngles Given: is parallel to D with transversal PQ. Explore: Measure the co-interior angles WX and WXD. How do I...? alculate the sum of these two angles. Drag points and X. W onclude: What do you notice about the sum of the co-interior angles WX and WXD? How do I...? Reset Pattern P ngle Measures in X D...measure an angle or side? m = construct a polygon interior m = 43 m = measure an area or perimeter?... create a table?... make a calculation? More... Explain why this relationship is called the -pattern. Q...measure an angle? Main Menu There are two "-patterns" in this diagram. an you find them? Help To measure : 1. Deselect. 2. Select point, then, then. Order is important! The second point chosen is the vertex. 3. Under Measure choose ngle.....measure the length of a side? Main Menu To measure side : 1. Deselect. 2. Select the two endpoints ( and ). 3. Under Measure choose Distance. Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 16

17 8.6: What s So Special? Guide Sheet Explore! Drag each vertex in the figure. s you drag vertices, look for some of the following: measurements that always seem to be equal to each other measurements that never seem to change measurements that might have a constant ratio (proportional) lines that always seem to be parallel or perpendicular line segments that always seem to be bisected figures that always seem to be congruent objects that don t seem to be connected, yet they move together when something is dragged Make an Hypothesis Decide which measurements you need to test your hypothesis. Drag each vertex again while you pay close attention to the way the object moves and to the way the measurements change. Test Your Hypothesis ollect and record evidence to test your hypothesis. What can you measure? angles lengths areas perimeters slopes What can you calculate? sums ratios formulas Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 17

18 Guiding Questions Examine the angles re any of the angles equal? Do any of the angle measures always add to give the same total? Does the measure of any angle always stay the same? re any of the angles cut in half (bisected)? Examine the line segments re any of the lengths equal? Is any length proportional with any other length? re any of the line segments cut in half (bisected)? Examine the lines re any of the lines parallel? re any of the lines perpendicular? re any of the slopes of the lines equal? Examine areas and perimeters re any of the areas equal? re any of the perimeters equal? re any of the shapes congruent? re any of the shapes similar? re any of the areas proportional? re any of the perimeters proportional? Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 18

19 What s So Special? (GSP 4 file) Special.gsp Investigation 1: Special Triangles? Hide Instructions K How do I...? M What's a Special Triangle? E I Exploring: Drag each vertex of every triangle. Hypothesizing: Make a hypothesis about each triangle. re any of the triangles special? L F D H G ollecting Evidence: What evidence can you find to support your hypothesis? What are you going to measure? calculate? tabulate? Making a onclusion: What kinds of triangles are shown? Were your hypotheses correct? Justify your answers using the evidence you collected. Main Menu Next page Investigation 2: Parallel or Perpendicular? Investigation 3: isected? Hide Instructions How do I...? T J Hide Instructions How do I...? Figure 1 Figure 2 Figure 3 Exploring: Drag the endpoints of each line segment. Hypothesizing: Make a hypothesis based on your exploration. ollecting Evidence: What evidence can you find to support your hypothesis? What are you going to measure? calculate? tabulate? Making a onclusion: What is special about some of the line segments? Were your hypotheses correct? Justify your answer using the evidence you collected. N P R S U O Q D L M K Main Menu Next page Exploring: Explore all three figures. Do you notice anything special? Record your observations. Hypothesizing: Make a hypothesis ollecting Evidence: What evidence can you find to support your hypothesis? What are you going to measure? calculate? tabulate? Making a onclusion: What is special about the intersecting line segments? Were your hypotheses correct? Justify your answer using the evidence you collected. O D Y W V X Z P R T Q S Main Menu Next page Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 19

20 What s So Special? (GSP 4 file) (continued) Investigation 4: Proportional? Show GSP Hint Investigation 5: Special Quadrilaterals? Hide Instructions How do I...? Exploring: Drag each vertex. Do you notice anything special? Measure the area and the perimeter of figures EFG and D. G F Hide Instructions How do I...? Exploring: Drag each vertex of every quadrilateral. Hypothesizing: Make a hypothesis about each quadrilateral. re any of the quadrilaterals special? D E F G Square Trapezoid Parallelogram Rectangle Hypothesizing: Is there anything special in this diagram? ollecting Evidence: What evidence can you find to support your hypothesis? What are you going to measure? calculate? tabulate? Making a onclusion: Was your hypothesis correct? Is anything in the diagram proportional? Justify your answers using the evidence you collected. E D ollecting Evidence: What evidence can you find to support your hypothesis? What are you going to measure? calculate? tabulate? Making a onclusion: What kinds of quadrilaterals are shown? Were your hypotheses correct? (Hint: Match the blue names to the quadrilaterals so that each name is used exactly once.) Justify your answers using the evidence you collected. Rule: Do not drag any of the shapes into a "non-quadrilateral" shape like this one. Rhombus Kite Quadrilateral Main Menu How do I...?...measure an angle or side?... construct a polygon interior... measure an area or perimeter? ngle Measures in m = 62 m = 43 m = 76 Show Show Defintions cute : triangle in which each of the three interior angles is acute. Obtuse : triangle containing one obtuse angle.... create a table?... make a calculation? Show Right : triangle containing a 90 angle....make a calculation? Show Equilateral : triangle with all sides equal. Main Menu To calculate the sum of the interior angles: 1. Deselect. 2. Under Measure choose alculate. 3. Select the measurement m (in the upper right corner) then the + sign (on the calculator) then the measurement m, then the + sign, then the measurement m, then OK Summary: Select measurements in the sketch and operations from the calculator and press OK at the end. Show Show Isosceles Scalene : triangle with two equal sides. : triangle with no equal sides. Perpendicular : Intersecting at 90. Show Parallel Lines: Lines in the same plane that do not intersect. They are always the same distance apart. Show isect: Means to cut in half. ngles or lengths can be bisected. Show ngle isector: line that divides an angle into two equal parts. Show Right isector of a line segment: line that is perpendicular to line segment and divides the line segment into two equal parts. Proportional: statement that two ratios are equal. (Two objects have exactly the same shape but are different sizes) Show Show Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 20

21 8.7: Learn the Lingo 1. Part a) shows an example of how to complete a word chart. omplete the remaining word charts. a) b) Term: Visual Representation: Term: Visual Representation: Equilateral Triangle Triangle Definition: n equilateral triangle is a triangle for which all sides have the same length. ssociation: Yield sign Definition: ssociation: c) d) Term: Visual Representation: Term: Visual Representation: Exterior ngle Interior ngle Definition: ssociation: Definition: ssociation: Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 21

22 e) f) Term: Visual Representation: Term: Visual Representation: Parallel Lines Transversal Definition: ssociation: Definition: ssociation: g) h) Term: Visual Representation: Term: Visual Representation: Perpendicular isector Diagonal Definition: ssociation: Definition: ssociation: Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 22

23 8.8: ngle pplications Determine the unknown angle. Give reasons for your answer. a) b) c) = = = DEG = d) e) f) OD = 64 o FOE = OF = O = 43 o OE = EOD = g) h) i) reate your own question! VRW = 42 SRT = 19 RSZ = o RT = RU o WX = WY YWX = 118 WXZ = o Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 23

24 8.9: Interior ngle Sums 1. omplete the chart. Diagram Number of sides Sum of interior angles Understanding The sum of the angles in any triangle is 180 o. 4 5 n 2. a) Determine the sum of the interior angles in a polygon with 15 sides. Show your work. b) Determine the number of sides in a polygon if the sum of the interior angles is Show your work. Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 24

25 3. Derek is building a deck for his summer job in the shape of a regular octagon. a) Define: regular octagon? b) Determine the measure of the interior angles of the deck. Show your work. 4. anadian $1 coin, known as a loonie, is a regular polygon with 11 sides, called an undecagon. a) Define a regular polygon with 11 sides. b) Determine the sum of the interior angles of the loonie. c) What is the size of one of the interior angles? Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 25

26 Solving Simple Equations Review (teacher) Instructions for Teacher: 1. Split students up into groups of 2 or opy, cut and paper clip together the following 5 equations. Each group will get all 5 equations. 3. Students will put each equation in its correct order. 4. Take up the solutions with students as a review leading into the next lesson. 1. 2aa 90 3a 90 3a a 30 Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 26

27 2. 2b b 110 2b b b3x 100 5x 100 5x x 20 Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 27

28 4. 4c c c 120 4c c g g g 140 5g g 28 Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 28

29 8.10: onnecting lgebra to Geometry 1. a) The sum of the interior angles in a triangle is: b) n equation that models the sum of the interior angles in this triangle is: c) Solve the equation to determine the value of x. d) Use the value of x to calculate the size of: W: Y: Z: 2. a) The sum of the angles in a right angle is: b) Write 2 equations to model the sums of the 2 sets of angles that add to 90º: (i) (ii) c) Solve these equations to determine the values. (i) solve for xº (ii) solve for yº d) Use the values of x and y to calculate the size of: P: Q: Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 29

30 3. Write an equation and solve for the unknown. State the theorem used to make the equation. a) b) c) d) e) f) g) h) Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 30

31 i) j) i j 2i 75 k) l) 2k 2l l k l m) n) 30 2n 80 2k 120 3n Modified TIPS4RM: Grade 9 pplied Unit 8: Plane Geometry 31

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