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G2 – Intro to Sketchpad Quadrilateral Definitions & Properties [Key Idea 1, 4, 7] The definition of a quadrilateral is a 4-sided planar figure. With only this definition, what properties can we determine? Exploration #1

Quadrilateral Angles Objective: To explore the angles of a quadrilateral and determine what type of quadrilateral has opposite ∠s ≅. a. Construct a quadrilateral by placing four points on the screen using the Point tool. Place four points on the screen by moving to four locations and clicking the mouse button. b. Construct a quadrilateral by selecting the four points in order. Now under the Construct menu, choose Segments. c. Measure the angles of the quadrilateral, using the three vertices that name each angle. Use the Measure menu and Calculate to find the sum of the angles. m ! ABC = 49.75° m ! BCD = 131.46°

D

m ! CDA = 60.60° m ! DAB = 118.20°

C

m ! A B C + m ! B C D + m ! C D A + m ! DAB = 360.00°

A

m!ABC

m!BCD

m!CDA

m!DAB

m!ABC+m!BCD+m!CDA+m!DAB

66.88°

114.62°

143.38°

35.12°

360.00°

48.82°

131.46°

143.38°

36.34°

360.00°

49.75°

131.46°

60.60°

118.20°

360.00°

B

d. Create a table for these measures. Select the four angle measures and the sum. Under the Graph menu, choose Tabulate. e. Change the size and shape of the quadrilateral. Point at one of the vertices, hold down the mouse button and drag the point on the screen. Release the button. The values in the last line of the table will change dynamically to reflect the most recent measurements. To record a new set of measurements, insert a new line into the table by double-clicking on the table. Repeat this several times to record a sequence of measurements. What stayed the same? Make a conjecture? f. Can you find a counter-example? That is, can you drag a vertex of the quadrilateral such that the sum of the angles is not 360˚? Can you explain why?

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g. Grab the vertices and move them around until m∠DAB = m∠BCD and m∠ABC = m∠CDA. This will be rather difficult, but do the best you can. Do NOT change them all to right angles. Your polygon should look something like the one below. What do you notice about the shape of ABCD? What is it getting close to? m ! ABC m ! BCD m ! CDA m ! DAB

= = = =

118.24° 61.76° 118.24° 61.76°

D

A

C

B

m ! A B C + m ! B C D + m ! C D A + m ! DAB = 360.00° m!ABC

m!BCD

m!CDA

m!DAB

m!ABC+m!BCD+m!CDA+m!DAB

66.88°

114.62°

143.38°

35.12°

360.00°

48.82°

131.46°

143.38°

36.34°

360.00°

49.75°

131.46°

60.60°

118.20°

360.00°

118.24°

61.76°

118.24°

61.76°

360.00°

h. Make a conjecture. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a . (Note: this is the converse of a familiar theorem that we often prove first: If quadrilateral ABCD is a then both pairs of opposite angles are This little demonstration, however, is a much more convincing argument than the usual one given by proving triangles congruent.

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Exploration #2

Constructing A Parallelogram Using The Definition Objective: To construct a parallelogram using opposite sides parallel and then to find the properties of parallelograms through investigation. a Construct the two sides of the parallelogram by constructing two segments with a common endpoint. Label the segments AB and BC . b. Select point C and segment AB, under the Construct menu choose Parallel Line. Select point A and segment BC, then under Construct choose Parallel Line. c. Place a point at the intersection of the two constructed parallel lines. This will label

D

C

B

A

the point of intersection, D. d. We want to define the parallelogram and get rid of the rest, so hide lines DC and AD . Construct segments DC and AD . Measure segments and angles. Make a conjecture about the sides of a parallelogram. Make a conjecture about the vertex angles of a parallelogram. e. Draw diagonals AC and BD . Label the intersection. Measure the segments and angles associated with the diagonals. D

C E

A

B

Make a conjecture about the diagonals of a parallelogram. What kinds of angles are formed at the intersection of the diagonals? Which angles are equal? What kind of angles do the diagonals make with each side of the parallelogram? (Such as ∠CDB, ∠CAB, etc.) Which angles are equal?

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f. The diagonals make 8 different triangles, such as ∆DCE, ∆DCB, etc. Based on your measurements, make some conjectures about these triangles. g. Move point B so that AB = BC. Now what is true? What is this shape called? What happened to ∠AEB and ∠BEC? Make a conjecture. A _____________ is a special case of the parallelogram; therefore, all _____________ are also parallelograms. In a _________________, the diagonals h. Move point B back so that AB ≠ BC. Move point C so that m∠ABC = 90˚. Now what is true? What is this shape? What happened to ∠BCD, ∠CDA, and ∠DAB? What happened to AC and BD ? Make a conjecture. A _____________ is a special case of the parallelogram; therefore, all _____________ are also parallelograms. In a _________________, the diagonals i. Move point B so that AB = BC and m∠ABC = 90˚. Now what is true? What is the shape? Make a conjecture. A _____________ is a special case of the parallelogram; therefore, all _____________ are also parallelograms. A _____________ is also a special case of the _____________ and the ______________; therefore, all _____________ are also __________________ and ___________________. In a ______________, the diagonals

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Exploration #3

Converses of Parallelogram Theorems We saw from Exploration 1 that the converse of the theorem that the opposite angles of parallelograms are congruent is also true. That is, if a quadrilateral has opposite angles congruent, then it is a parallelogram. Objective: To determine through constructions what other converses are true for parallelograms. 1. Constructing a quadrilateral with opposite sides congruent: a. Construct the two sides of the quadrilateral, AB and BC . b. Select point C and segment AB and construct a Circle by Center+Radius. This locates all points of B distance AB from C. c. Select point A and C segment BC and do the same. A d. Label the point at the intersection of the two circles that is on the same D side of line AB as C. The distance from D to C = AB and the distance from D to A = BC. e. Construct segments CD and AD. You now have a quadrilateral with opposite sides congruent. f. Determine if this is a parallelogram by measuring the slopes of the opposite sides. (Note: If we had selected the other point of intersection of the circles, we would have a quadrilateral with opposite sides congruent, but CD would cross AB making a non-convex figure.) 2. Constructing a quadrilateral with one pair of opposite sides both parallel and congruent. a. Construct the two sides of the quadrilateral, AB and BC . b. Construct a line through point C parallel to AB . c. Construct a line through point C equal in length to AB by constructing a circle with radius = AB and center C. d. Label point D where the circle and B parallel line intersect on the same side of BC as A. C

A

D

e. Construct segment AD; hide the circle. You now have a quadrilateral with AB and CD both parallel and equal in length. Determine if this is a parallelogram by measuring the slopes of the opposite

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sides. (Note: If we had selected the other point of intersection of the circle and the parallel line, we would have a quadrilateral with one pair of opposite sides both parallel and congruent, but AD would cross CB making a non-convex figure.) 3.

Constructing a quadrilateral with diagonals that bisect each other. a. Construct a segment. Now construct a Midpoint. While the midpoint is selected, under the Transform menu choose Mark Center. b. Place a point on the screen, not on the segment. Select the point and under the Transform menu choose Rotate. In the dialog box enter 180°. c. Label one endpoint of the segment A, the B new point B, the other end of the segment C, and the reflected point D. (To change the labels, C double-click on the label with the text tool.) d. Construct a segment between B and D. e. You have now constructed BE = ED. Select E the points in order starting at A and going clockwise (ABCD in my diagram) and under the A Construct menu choose Segments to construct quadrilateral ABCD. f. Determine if this is a parallelogram by D measuring the slopes of the opposite sides.

What other conjectures about parallelograms might we test through constructions? Exploration #4

Trapezoids The definition of a trapezoid is a quadrilateral with one pair of opposite sides parallel. If we define a trapezoid as having exactly one pair of opposite sides parallel, then it has no significant subsets. If we define a trapezoid as having at least one pair of opposite sides parallel, then the set of parallelograms is a subset of trapezoids. 1.

Objective: To construct a trapezoid using the definition. a. Construct the two sides of the quadrilateral, AB and BC . b. Through point C, construct a line parallel to AB . c. Place a point D anywhere on the parallel line on the same side of BC as A. C

D

A

d. Construct segment AD.

B

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D B A

(Note: Both of the above are possible configurations for trapezoid ABCD. If D is positioned such that AD is parallel to BC , we then have a parallelogram. That is why in some textbooks parallelograms are categorized as special cases of trapezoids. This parallels our previous examples for parallelograms: • a square is a special case of a rectangle because a rectangle can be dragged to form a square; • a rectangle is a special case of a parallelogram because a parallelogram can be dragged to form a rectangle; and • a rhombus is a special case of a parallelogram because a parallelogram can be dragged to form a rhombus. By the same logical argument, using inclusive definitions, a parallelogram should be a special case of the trapezoid because a trapezoid can be dragged to form a parallelogram.) 2.

Objective: To construct the isosceles trapezoid using the definition. a. Construct the two sides of the quadrilateral, AB and BC . b. Through point C, construct a line parallel to AB . c. Select point A and segment BC and construct a Circle by Center+Radius. D

C

A

B

d. Construct a point D at the intersection of the circle and the parallel line farthest from point C. e. Construct segment AD. (Note: If we had selected the other point of intersection, we would have a parallelogram. This illustrates why having one pair of opposite sides parallel and the other pair of opposite sides congruent is a necessary condition for a parallelogram, but not a sufficient condition.) f. Construct the diagonals of this trapezoid and make some conjectures. Determine which converses are also true.

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Exploration #5

Kites A kite is defined as a quadrilateral with two pair of adjacent sides congruent. Objective: To construct a kite using the definition. a. Construct a segment AB. b. Select point B and then point A, and under Construct choose Circle by Center+Point. c. While the circle is selected, under Construct choose Point on Circle. A d. Construct segment BC. Hide the circle. B e. From C draw any segment. Select C and the segment and under Construct choose Circle by Center+Radius. f. Select A and the segment and under Construct choose Circle by Center+Radius. (See construction below.) C g. Construct a point at the intersection of the two circles, interior to ∠ABC. If the circles do not intersect, adjust the length of the segment. Label the intersection point D. Hide the circles and the segment from step e. A h. Measure the angles of kite ABCD. Make a conjecture. i. Construct the diagonals AC and BD . Construct a point at the intersection. Measure the diagonals, segments, and angles. Make some conjectures. (See construction on next page.) A

D

B E

D C

j. Drag vertices of the kite. What other quadrilaterals can you make? What quadrilaterals can be classified as subsets of the family of kites?

G8 – Sketchpad Constructions 1. Copy AB and CD in the space below.

C

B

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A

C

B

•

Copy angle 1 and angle 2 in the space below.

2

1

•

Bisect the given angles.

D

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Bisect AB andCD .

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A

D

C

B

•

Construct a line perpendicular to ST that passes through the point P.

T

S

P

•

Construct a line perpendicular to QR that passes through the point P.

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Q

P

R

6.

Draw a line parallel to EF .

F

E

7.

Construct an equilateral ∆ABC, use the given line-segment below.

Summer 2006 I2T2

Geometry

C

B

8.

Construct the center of the circle below.

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Summer 2006 I2T2 9.

Geometry

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Inscribe a circle in the triangles below.

E

G F

X

Z

Y

10.

Construct a regular hexagon.

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