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Geometry Geometers’ Sketchpad Properties of Quadrilaterals Name___________________________________ Open file in Class_Pickup folder: “Properties of quadrilaterals.GSP” There are 2 tabs at the bottom, “6.4 parallelograms” and “6.5 Trapezoids and kites” A higher accuracy decimal place may be required to recognize some of these patterns. To change the units preference, go to Edit->Preferences. Changing Distance to “thousands” may help you see the patterns. Discovery Activity 1: Investigating properties of special parallelograms….rhombus, rectangle, square Part I. Create the parallelogram 1. Select tab “6.4 parallelograms” 2. Create a parallelogram ABCD using the segments AB and BC given in the diagram. a. Construct a line through C parallel to AB i. Select AB and point C. “Construct->Parallel lines” b. Construct a line through A parallel to BC. i. Select BC and point A. “Construct->Parallel lines” c. Label the point of intersection of these 2 lines as D. d. Hide the parallel lines i. Select the 2 parallel lines (make sure point D is not selected) ii. “Display->Hide Parallel Lines” e. Construct AD and CD to complete the parallelogram ABCD f. Construct the diagonals of parallelogram ABCD. i. Label the point of intersection as E. Part II. Set up measurements to be used for the investigation. M1: Angle measure at a single vertex (to be used for checking for rectangle): mABC M2: Length of 2 adjacent sides (to be used for checking for rhombus): AB and BC M3: Angle measure at the intersection of the diagonals: mAEB M4: Length of each diagonal AC and BD M5: Angle measure of the two angles at a vertex created by the diagonal. mADE and mCDE Part III. Investigate and make conjectures 1.Rectangles a. Drag vertex B to make the parallelogram into a rectangle (look at M1). Why is only 1 angle measurement sufficient to verify we have a rectangle? ________________________________________________________ b. Record your measurements for M4: AC =______________ BD =___________________________ c. Conjecture 1: The diagonals of a rectangle_______________________________________________________ 2. Rhombus a. Drag a vertex to make a rhombus (look at M2) Record your measurements for M3: mAEB =______________ and M5: mADE =___________ and mCDE =______________________ b. Conjecture 2: The diagonals of a rhombus____________________________________________________ and __________________________________________________________________. 3. Square a. Drag a vertex to make a square (look at M1 & M2). Record your measurements for M3: mAEB =__________, M4: AC =_____________ BD =____________ M5: mADE =___________ and mCDE =____________ b. Conjecture 3: The diagonals of a square________________________________________________________ and ____________________________________________________________________________ and _____________________________________________________________________________. Part IV. Extension. Do you think the converses of your conjectures are true? Try manipulating the diagonals to verify one of your conjectures and give your conclusion here.______________________________________________________ Discovery Activity 2: Investigating properties of kites and trapezoids Using the current file “Properties of quadrilaterals.GSP”, Part I. Create a Trapezoid 1. Select tab “6.5 Trapezoids”. Create a trapezoid ABCD using the segments AB and BC given in the diagram. 2. Construct a line through C, parallel to AB a. Select AB and point C. “Construct->Parallel lines” 3. Construct a segment from A to a point on the parallel line. a. Label the point D 4. Hide the parallel line used in the construction a. Select the parallel line (make sure nothing else is selected) b. “Display->hide Parallel line” 5. Construct CD to complete the trapezoid ABCD. 6. Construct the diagonals of the trapezoid. Part II. Set up measurements for the investigation. M6: Length of both legs of the trapezoid: BC and AD M7: Measure each bottom base angle: mDAB , mABC , and and measure of each top base angle : mBCD , m CDA M8: Length of each diagonal AC and BD Part III. Investigate and make conjectures for an isosceles trapezoid 1. Drag point D until you have an isosceles trapezoid. (use M6 to verify). (Observe the changes in measurements M7 and M8 as you drag point D) 2. Record measurements for M7: Bottom base angles: mDAB = _______ and mABC =_______________ Top base angles: mBCD = _______ and mCDA =______________ 2. Record your measurements for M8: AC = ______________ and BD=____________________ 3. Make a conjecture Conjecture 1: The base angles of an isosceles trapezoid _________________________________ Conjecture 2: The diagonals of an isosceles trapezoid ___________________________________ Part IV. Extension: Create a new sketch and construct a kite (no parallel lines needed for this one). Make a conjecture about the angles formed at the intersection of the diagonals of the kite. _____________________________________________________________________________