The quadrilateral family is organized according to the number pairs of sides parallel in a particular quadrilateral. Given a quadrilateral, there are three distinct possibilities: both pairs of opposite sides are parallel, one pair of opposite sides is parallel, and neither pair of opposite sides is parallel. As you do this activity, think about what the family tree of quadrilaterals would look like. 1. The figure below is a type of quadrilateral called a parallelogram; a figure in which both pairs of opposite sides are parallel. Use patty paper to trace and label the parallelogram. A B D C a. Use your patty paper tracing to compare the measures of A, B, C, and D. What appears to be true? Summarize your findings below. b. Use your paper tracing to compare the side lengths of the parallelogram. What appears to be true? Summarize your findings below. c. Pick a pair of consecutive angles from the parallelogram. Find the sum of their measures. Record your answer below. Pick a second pair of consecutive angles from the parallelogram. Find the sum of their measures. Record your answer below. Write a conjecture about any pair of consecutive angles in a parallelogram. d. Use a straightedge to draw the diagonals on the parallelogram on your patty paper. Fold your paper so that point A lies directly on top of point C, and then again so that point B lies directly on top of point D. What do the creases in your paper lead you to believe about the diagonals of the parallelogram? 2012, TESCCC 09/10/12 page 1 of 7
2. The figure below is a rectangle. A rectangle is a type of quadrilateral that has four right angles. H I K J a. Use your pencil and a straightedge to draw Line HI, Line KJ, and Line HK. Let Line HK be a transversal for the other two. What conclusions can you make about Line HI and Line KJ based on your knowledge of lines and transversals? Explain. b. Use your pencil and a straightedge to draw Line KJ. Let Line KJ be a transversal for Line HK and Line IJ. What conclusions can you make about Line HK and Line IJ based on your knowledge of lines and transversals? Explain. c. What do your findings from part a and b lead you to believe about the rectangle? d. Use a straightedge and your pencil to draw the diagonals of the rectangle. Use patty paper tracings to compare the lengths of the two diagonals. What appears to be true? 2012, TESCCC 09/10/12 page 2 of 7
3. The figure below is a rhombus. A rhombus is a type of quadrilateral that has four congruent sides. P S Q R a. Find the measure of the angles of the rhombus. Record your answers below. b. Let Line SR be a transversal for Line SP and Line RQ. What conclusions can you make about Line SP and Line RQ based on your knowledge of lines and transversals? Explain. c. Let Line SP be a transversal for Line SR and Line PQ. What conclusions can you make about Line SR and Line PQ based on your knowledge of lines and transversals? Explain. d. What do your conclusions in parts b and c lead you to believe about the rhombus? Explain. e. Use a straightedge and your pencil to draw the diagonals of the rhombus. Measure the four angles created by the intersection of the diagonals. Record your answers below. Write a conjecture about the diagonals of a rhombus. 2012, TESCCC 09/10/12 page 3 of 7
4. The figure below is a square. A square is a type of quadrilateral that has four right angles and four congruent sides. W X Z Y a. Based on the previous explorations, what conclusions can you make about the square? Explain your reasoning. 5. Based on the previous explorations, what similarities did you discover about the parallelogram, rectangle, rhombus, and square? 2012, TESCCC 09/10/12 page 4 of 7
6. The figure below is a trapezoid. A trapezoid is a type of quadrilateral with exactly one pair of parallel sides. A B D C a. Use a protractor to find the measures of the angles of the trapezoid. Record your answers in the figure. Use this information to justify a pair of parallel sides in the figure above. Explain your reasoning. b. Using a piece of patty paper and a straightedge, trace the figure above and label the vertices. Fold the patty paper so that the parallel sides coincide. Draw a line segment along the crease of your paper. Label the segment MN. c. MN is called the median for the trapezoid. What is true about points M and N? d. Find the length of MN, AB, and DC to the nearest tenth of a centimeter. Find the sum of AB and DC, and divide by two. How does your result compare to MN?. e. Based on your answer to part d, what conjecture can you write about the median of a trapezoid? 2012, TESCCC 09/10/12 page 5 of 7
7. The figure below is a special type of trapezoid called an Isosceles Trapezoid. An Isosceles Trapezoid has exactly one pair of congruent sides. T R P A a. Use a straightedge and a piece of patty paper to trace the trapezoid. Label the vertices on your tracing. Fold and crease the patty paper so that point T and R coincide. b. Based on your folding, what can you conclude about the parts of the trapezoid? c. Based on your findings in part a and b, write a conjecture about isosceles trapezoids. 2012, TESCCC 09/10/12 page 6 of 7
8. The figure below is a type of quadrilateral called a Trapezium. A trapezium is a quadrilateral with no pairs of parallel sides. P Q R S a. A special type of trapezium is a kite. A kite is a trapezium with two pairs of congruent adjacent sides. The figure below is a kite. K E I T b. Use a piece of patty paper and a straightedge to trace the kite. Label the vertices of the kite. Use a straightedge to draw the diagonals of the kite on your patty paper tracing. c. Use a protractor to measure the angles formed by the intersection of the diagonals. Record the measures on your patty paper tracing. What conclusion can you make about the diagonals of the kite? d. Fold and crease your patty paper tracing so that E coincides with I. Examine the crease in your paper. What does this verify about diagonal EI? 2012, TESCCC 09/10/12 page 7 of 7