Investigating Rotational & Reflectional Symmetry of Quadrilaterals

1Investigating Rotational & ReflectionalSymmetry of QuadrilateralsTask 1 – Investigating Rotational Symmetry• Open the Sketchpad sketch entitled “QuadsNCTM.”• In the Definitions file of the sketch, check out the related terms and theirdescriptions.• In the Rotational Symmetry file of the sketch, follow the directions of thesketch. Record your findings in the table below.• In the Kite file of the sketch, follow the directions of the sketch. Record yourfindings in the table below.• In the Trapezoid file of the sketch, follow the directions of the sketch. Recordyour findings in the table below.Task 2 - Investigating Reflectional Symmetry• In the Midsegments & Symmetry file of the sketch, follow the directions of thesketch. Record your findings in the table below.• In the Diagonals & Symmetry file of the sketch, follow the directions of thesketch. Record your findings in the table belowType ofSymmetryParallelogram Rhombus Rectangle Square Trapezoid IsoscelesTrapezoidKiteRotationalSymmetry?Angle(s) ofSymmetryReflectionalSymmetry?Mid-midline(s) –MD or IT?Diagonal(s)– QA orUD?• In the Quads file of the sketch, follow the directions of the sketch. Recordyour findings in the table.

2Quadrilateral Parallelogram Rhombus Rectangle Square Trapezoid IsoscelesTrapezoidKiteOpposite sides≅ ?Oppositeangles ≅ ?Opposite sidesparallel?At least onepair of parallelsides?At least oneright angle?Consecutiveanglessupplementary?Diagonals ≅ ?Diagonals ⊥ ?Diagonalsbisect eachother?Diagonalsbisect anglesQ, U, A & D?Type ofVarignonQuadrilateral*CyclicQuadrilateral?Diacenter?Medcenter?

Task 3 – Varignon Quadrilaterals• In the Varignon quad file of the sketch, follow the directions of the sketch.Record your findings in the table.Task 4 – Making Quadrilaterals from Triangle Pairs• Open the Triangles & Quads Sketchpad sketch.• In the Generic Triangle Maker file of the sketch, follow the directions of thesketch. Record your findings in the table below; what type of quadrilateral isformed when ∆ABC is reflected? When ∆ABC is rotated? Complete the gridwith: isosceles triangle, parallelogram, rhombus, rectangle, square, kiteor concave kite.• In the Obtuse Triangle Maker file of the sketch, follow the directions of thesketch. Record your findings in the table below.• In the Right Triangle Maker file of the sketch, follow the directions of thesketch. Record your findings in the table below.• In the Isosceles Maker file of the sketch, follow the directions of the sketch.Record your findings in the table below.• In the Equilateral Triangle Maker file of the sketch, follow the directions of thesketch. Record your findings in the table below.GenericTriangleMakerObtuseTriangleMakerRight TriangleMakerIsoscelesTriangleMakerEquilateralTriangleMakerReflection inACReflection inABReflection inBC180° Rotationabout midpointof AC180° Rotationaboutmidpoint ofAB180° Rotationaboutmidpoint ofBC

4Theorem: The vertices of two congruent triangles that share a common sideform a parallelogram, a kite or a cyclic quadrilateral.Task 5 – A Closer look at the Quadrilaterals• In the Parallelogram file of the sketch, follow the directions of the sketch.Complete the following.o A quadrilateral is a parallelogram if any of the following exist:••••o In a parallelogram:•••••• In the Isosceles Trapezoid file of the sketch, follow the directions of thesketch. Complete the following.o Isosceles trapezoid theorems:•••••••• In the Rectangle file of the sketch, follow the directions of the sketch.Complete the following.o A rectangle is a special case of a(n)________________.o A quadrilateral is a rectangle if any of the following conditions exist:•••o A rectangle is a _______________ .o In a rectangle:••••• In the Rhombus file of the sketch, follow the directions of the sketch.Complete the following.o Rhombus theorems:•••

5••• In the Definitions file of the sketch, activate the diagonals button. Completethe following.o A quadrilateral is a kite if and only if• It satisfies its definition – a kite is a quadrilateral in which atleast one diagonal defines a line of symmetry oralternatively, a kite is a quadrilateral with two distinct pairs ofconsecutive, congruent sides.• One of its diagonals is the __________________________ .• One of its diagonals is the ____________ of the angles at itsendpoints.o The bisectors of the angles of a kite are concurrent.Assessment – Quadrilateral Riddles – Who Am I?1. No matter how I change, I always have two pairs of sides with thesame length and at least one line of symmetry. Sometimes I have onlyone line of symmetry. Which quadrilateral am I?2. No matter how I change, I always have at least one pair of parallelsides. Which quadrilateral am I?3. Whenever I have a right angle, I’m a rectangle. Sometimes not all ofmy sides have the same length and I have no right angles. Whichquadrilateral am I?4. I’m always a parallelogram. I’m always a kite. Sometimes I’m not allright. Which quadrilateral am I?5. I always have at least two lines of symmetry. Sometimes not all of mysides have the same length. Which quadrilateral am I?6. I can have two right angles or four right angles, but never just one.Which quadrilateral am I?7. I always have the measures of any two adjacent anglessupplementary. Sometimes not all of my sides have the same length.I don’t always have right angles. Which quadrilateral am I?8. Sometimes I have four different side lengths. Sometimes no pair of myangles is supplementary. Which quadrilateral am I?

6Task 6 – Quadrilateral Constructions Using Symmetry• How can a parallelogram be constructed using its symmetryand transformations? Describe a method.• How can a kite be constructed using its symmetry andtransformations? Describe a method.• How can an isosceles trapezoid be constructed using itssymmetry and transformations? Describe a method.• How can a rectangle be constructed using its symmetry andtransformations? Describe a method.• How can a rhombus be constructed using its symmetry andtransformations? Describe a method.• How can a square be constructed using its symmetry andtransformations? Describe a method.* It might be helpful to discuss the notion of angle-side duality. For example,the rectangle and rhombus are duals – the rectangle’s angles are all equaland the rhombus’s sides are all equal. Also, duality of the kite and theisosceles trapezoid is illustrated in the table:Isosceles TrapezoidKite• Two pairs of equal adjacent angles • Two pairs of equal adjacent sides• One pair of equal opposite sides • One pair of equal opposite angles• Circumscribed circle (cyclic) • Inscribed circle (circum quad)• An axis of symmetry through one pairof opposite sides• An axis of symmetry through one pairof opposite angles

7ReferencesBattista Michael T. 1998. Shape Makers – Developing Geometric Reasoningwith the Geometer’s Sketchpad®. Emeryville: Key Curriculum Press.De Villiers, Michael D. 1999. Rethinking Proof with the Geometer’sSketchpad®. Emeryville: Key Curriculum Press.Keyton, Michael. 1996. 92 Geometric Explorations on the TI-92. Dallas: St.Mark’s School.Wells, David. 1991. The Penguin Dictionary of Curious and InterestingGeometry. England: Penguin Books.