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Properties of Quadrilaterals Question: How do the properties of quadrilaterals help in defining and constructing special quadrilaterals? How are the members of the quadrilateral related? Launch Students in Ms. Fox’s class are studying properties of quadrilaterals and how to define and construct them. Today they are discussing isosceles trapezoids. Mary claimed that all you need to do to construct an isosceles trapezoid is to make sure that one pair of sides are parallel. George said that you just need to have two opposite sides congruent. Marla said that you just need to make sure that two pairs of angles are congruent. Michael said that you just need to have congruent diagonals. Wendy said that you only need to be sure that consecutive angles are supplementary. 1. Explain why each student’s statement is partially true, but provide a counter example to show why it is not always true. 2. Write a definition of an isosceles trapezoid. Note: A definition in mathematics should contain only sufficient and necessary information. 10/08/06 1 Draft Investigation: Can you name these special quadrilaterals? 10/08/06 2 Draft 1. Complete the table below by measuring the quadrilaterals on the previous page to determine which properties are true for each kind of quadrilateral. Properties of Quadrilaterals Parallelogram Rhombus Rectangle Square Isosceles Trapezoid Trapezoid Kite Exactly 1 pair of parallel sides 2 pairs of parallel sides Exactly 1 pair of congruent sides 2 pairs of congruent sides All sides congruent Perpendicular diagonals Congruent diagonals Diagonals bisect angles Diagonals bisect each other 2. In the chart above you verified the properties of quadrilaterals by measuring parts. How could you prove the following without measuring? Given: Rectangle FLAG. F L G A Prove: FA LG 3. Three vertices of a rectangle have coordinates (3,0), (-1, 5), and (-5,2). Find the coordinates of the fourth vertex. Verify that the figure is a rectangle. 10/08/06 3 Draft 4. Which of the following would be sufficient to prove that a quadrilateral is a parallelogram? If not sufficient, provide a counterexample to illustrate. a. Prove that both pairs of opposite sides are parallel. b. Prove that a pair of adjacent angles are supplementary. c. Prove that both pairs of opposite sides are congruentl. d. Prove that the sum of the angles is always 360o. e. Prove that the diagonals bisect each other. f. Prove that one pair of opposite sides are both parallel and congruent. 5. How could you prove or verify that the figure shown is a parallelogram? 8 C 6 D 4 2 B A 5 10 15 Conclusions: Based on what you have discovered about the properties of special quadrilaterals, write a definition of each figure. Compare your definitions with your classmates. 1. 2. 3. 4. 5. 6. 7. Parallelogram Rhombus Rectangle Square Trapezoid Isosceles Trapezoid Kite 10/08/06 4 Draft In Class Problems: Determine if each statement is Always, Sometimes, or Never True and explain how you know. 1. A rhombus is a parallelogram. 2. A square is a trapezoid. 3. A kite is a quadrilateral. 4. A parallelogram is a rectangle. 5. A square is a rectangle. 6. An isosceles trapezoid is a trapezoid. 7. A rectangle is a parallelogram. 8. A square is a rhombus. Draw a diagram to show the relationship among the members of the quadrilateral family. Closure: How do the properties of quadrilaterals help in defining and constructing special quadrilaterals? How are the members of the quadrilateral related? 10/08/06 5 Draft Homework: 1. The diagonals of a kite must be perpendicular to each other. Use this property to prove that the figure shown could be a kite. What else would you need to prove to show that this figure is a kite? I 6 K 4 2 -5 5 10 T E -2 -4 2. Determine the coordinates of point Z so that the figure WXYZ formed is a rectangle. Verify that the figure is a rectangle. W(4,-3), X(4,4), Y(-4,4), Z(___, ___) Determine the coordinates of point P so that the figure MNOP formed is an isosceles trapezoid. Verify that the figure is an isosceles trapezoid. M(-3,-2)), N(0,2), O(5,2), P(___, ___) 3. Michael claims that a good definition for an isosceles trapezoid is that it is a quadrilateral with exactly one line of symmetry. Can you think of a counterexample that would disprove his definition? 10/08/06 6 Draft 4. One diagonal of a rhombus lies on the line with equation y = -4x + 4. The 1 other diagonal lies on the line with equation y = x – 3. Two of the vertices 4 of the rhombus are (4,-2) and (0, 4). Find the coordinates of the other two vertices. 5. The median of an isosceles trapezoid is a segment joining the midpoints of the two congruent legs of the trapezoid. The length of the median is one-half the sum of the lengths of the parallel sides. Prove/verify this using the drawing below. 4 B C 2 -5 5 A D -2 -4 10/08/06 7 Draft C 6. The drawing to the right represents the construction of the perpendicular bisector of CD . What properties of quadrilaterals can be used to prove why this construction works? Use these properties to prove that the line drawn is the perpendicular bisector of CD . D 7. The following sketch is a general quadrilateral. Find the coordinates of the midpoints of each side. Connect the midpoints. What kind of figure is formed? How do you know? Do you think that joining the midpoints of the sides of any quadrilateral will always produce the same results? B C 8 6 4 A 2 -5 5 10 D 10/08/06 8 Draft 10/08/06 9 Draft