Quadrilaterals GETTING READY FOR INSTRUCTION


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1 Quadrilaterals / Mathematics Unit: 11 Lesson: 01 Duration: 7 days Lesson Synopsis: In this lesson students explore properties of quadrilaterals in a variety of ways including concrete modeling, patty paper tracings, measurement, and coordinate geometry. The explorations lead to discovery of the various properties of quadrilaterals that naturally group quadrilaterals according to the number of pairs of parallel sides. Students use properties to not only solve quadrilateral problems but justify types of quadrilateral relationships through proof and coordinate geometry. Students revisit quadrilateral groupings as they create a structure that illustrates the lineage of quadrilaterals. TEKS: G.2 Geometric structure The student analyzes geometric relationships in order to make and verify conjectures. G.2B Make conjectures about angles, lines, polygons, circles, and threedimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. G.5 Geometric patterns The student uses a variety of representations to describe geometric relationships and solve problems G.5B Use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles. G.7 Dimensionality and the geometry of location The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. G.7B Use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons. G.7C Derive and use formulas involving length, slope, and midpoint. GETTING READY FOR INSTRUCTION Performance Indicator(s): Investigate, compare, identify, and apply properties of quadrilaterals, including the trapezium, kite, trapezoid, parallelogram, rhombus, rectangle, and square in order to create a family tree of quadrilaterals and identify quadrilaterals by analyzing characteristics with coordinate proofs. (G2.B, G5.B, G.7BC) Key Understandings and Guiding Questions: Different types of quadrilaterals have specific characteristics and properties. What characteristics are used to identify types of quadrilaterals? What are the specific properties for each type of quadrilateral? What characteristic differentiates between a trapezium, a trapezoid, and a parallelogram? How do a rhombus, rectangle, or square relate to a parallelogram? Quadrilaterals can be identified by analyzing characteristics with coordinate proofs. What characteristics of a figure must be analyzed in a coordinate proof? How can the slope, midpoint, and distance formulas be used to analyze quadrilaterals with coordinate geometry? Vocabulary of Instruction: trapezium kite trapezoid isosceles trapezoid parallelogram rhombus rectangle square consecutive angles opposite sides consecutive sides opposite angles 2007, TESCCC Revised 10/01/07 page 1 of 50
2 Materials: colored paper patty paper protractor chart paper markers straightedge Resources: Advance Preparation: 1. Handout: Quadrilaterals and Angle Sums (one copy per student) 2. Handout: Quadrilaterals and Angle Sums Practice (one copy per student) 3. Handout: All in the Family (one copy per student) 4. Handout: Properties of Quadrilaterals (one copy per student) 5. Handout: Quadrilaterals and Coordinate (one copy per student) 6. Handout: Trace the Ancestry (one copy per student) Background Information: The explorations in this lesson rely on prior knowledge in a variety of forms including parallel lines and transversals, properties of triangles, and right triangle relationships in order to discover properties of quadrilaterals and develop a structure for quadrilateral lineage. In addition, students revisit much of the topics from the second unit, Functions in The Coordinate Plane, as they use slope, midpoint formula, and the distance formula to investigate quadrilaterals using coordinate geometry. INSTRUCTIONAL PROCEDURES Instructional Procedures ENGAGE 1. Assemble students into appropriate groups of 4 to 5 students each. 2. Have students create a concept map with the word quadrilateral as the beginning. 3. After about 5 minutes have groups post their concept maps on the walls of the classroom. 4. Have each group give an overview of the content of their concept map. EXPLORE 1 1. Distribute copies of Quadrilaterals and Angle Sums to each student. 2. Have students work through the activity with a partner or in a small group setting. Notes for Teacher NOTE: 1 Day = 50 minutes SUGGESTED DAY 1 MATERIALS chart paper markers TEACHER NOTE The purpose of the ENGAGE activity is to allow the students to recall prior knowledge of quadrilaterals, and to allow the teacher to preassess the level of understanding of the students. MATERIALS handout: Quadrilaterals and Angle Sums (one copy per student) protractor straightedge colored paper TEACHER NOTE The purpose of this EXPLORE activity is to allow students the opportunity to discover the angle sum relationship for quadrilaterals. Students model concretely the angle sum of a quadrilateral as they did in with triangles in a previous unit. 2007, TESCCC Revised 10/01/07 page 2 of 50
3 Instructional Procedures Notes for Teacher EXPLAIN 1 SUGGESTED DAY 2 1. Debrief the previous EXPLORE activity and facilitate a discussion of MATERIALS student findings using the questions below. Facilitation Questions: handout: Quadrilaterals and Angle Sums Practice (one copy per What are some ways to distinguish convex quadrilaterals from concave quadrilaterals? As the names imply, concave quadrilaterals are dented in while convex ones are not. Geometrically, both diagonals of a convex quadrilateral are contained within the interior of the quadrilateral (except the endpoints), while one of the diagonals of a concave quadrilateral has points on the exterior of the quadrilateral. student) Ask each group: What was the sum of the measure of the angles of the quadrilateral that you drew? 360º How did you verify your angle sum? Explain. By arranging the angles so that they all had the same vertex, the angles were observed to all be adjacent to one another; therefore, completing a circle about the point or 360º of rotation. What can you conclude about the sum of the measures of the angles of any convex quadrilateral? The sum is always 360º 2. Distribute Quadrilaterals and Angle Sums Practice to the students. 3. Have students complete Quadrilaterals and Angle Sums Practice. EXPLORE 2 SUGGESTED DAY 2 AND 3 1. Distribute copies of All in the Family to students. MATERIALS 2. Have students complete the All in the Family with a partner or in groups. handout: All In The Family (one copy per student) patty paper protractor straightedge TEACHER NOTE The purpose of this EXPLORE is to facilitate students in the discovery of properties of quadrilateral using patty paper tracings and their previous knowledge. EXPLAIN 2 SUGGESTED DAY 3 AND 4 1. Debrief the previous EXPLORE activity with students. Be sure to facilitate MATERIALS a discussion regarding common properties of certain quadrilaterals. handout: Properties of Facilitation Questions: Quadrilaterals (one copy per Which quadrilaterals had no pair of opposite sides parallel? The student) Trapezium and the Kite. Which quadrilaterals had exactly one pair of opposite sides parallel? The trapezoid and the isosceles trapezoid. Which quadrilaterals had both pairs of opposite sides parallel? The parallelogram, rectangle, rhombus, and square. TEACHER NOTE Which quadrilaterals had characteristics of a rectangle? The The Practice Problems from square. Properties of Quadrilaterals provide Which quadrilaterals had characteristics of a rhombus? The students with an opportunity to square. demonstrate their knowledge of Can you give examples of how can this information could be used quadrilaterals and their properties, and to classify quadrilaterals? Explain. Rectangles, rhombi, and squares albeit brief, demonstrate their ability to are all unique quadrilaterals yet they are related. Each has their own prove aspects of quadrilaterals based unique characteristics but they all have the characteristics of a on their properties. Should the teacher parallelogram. A square is a type of rectangle because it satisfies the wish to include more deductive proofs, a defining characteristics of a rectangle. etc. suggestion would be to resort to proving 2007, TESCCC Revised 10/01/07 page 3 of 50
4 Instructional Procedures 2. Distribute Properties of Quadrilaterals to students and clarify the properties for each type of quadrilateral. 3. Have students complete the Practice Problems form Properties of Quadrilaterals in order to demonstrate their understanding of quadrilaterals and their properties. Notes for Teacher any of the numerous properties discovered in the previous EXPLORE phase as time permits. ELABORATE SUGGESTED DAY 4, 5 AND 6 1. Distribute Quadrilaterals and Coordinate to students. MATERIALS 2. Have students complete Quadrilaterals and Coordinate. handout: Quadrilaterals and Coordinate (one copy per student) TEACHER NOTE The purpose of the ELABORATE phase is to give students the opportunity to explore and use properties to justify types of quadrilaterals using coordinate geometry. Students rely on previous knowledge of slope, midpoints, and the distance formula, Pythagorean Theorem, and in some cases trigonometric ratios. EVALUATE SUGGESTED DAY 7 1. Distribute Trace the Ancestry to students. MATERIALS 2. Have students complete Trace the Ancestry. handout: Trace the Ancestry (one copy per student) Teacher note: In the EVALUATE phase, students revisit properties of quadrilaterals to create a structure that demonstrates their understanding of quadrilateral relationships, and justify classifications of quadrilaterals by coordinate proof. 2007, TESCCC Revised 10/01/07 page 4 of 50
5 Quadrilaterals and Angle Sums (Key) Quadrilaterals are four (quad) sided (lateral) geometric figures. Quadrilaterals can be convex or concave. CONVEX CONCAVE Investigation: Angles in a convex quadrilateral 1. Draw a foursided convex quadrilateral on a sheet of colored paper. See students samples 2. Measure the four angles to the nearest degree. Record your measures below. See students samples 3. Find the sum of the measure of the four angles. See students samples 4. Tear off the four angles and arrange them around the point below. See students samples 5. What conjecture can be made about the sum of the measures of the four angles of a convex quadrilateral? The sum of the measures of the angles of a convex polygon is 360º. 6. Would this conjecture hold true for the sum of the angles of a concave quadrilateral? No, one of the interior angles would be greater than 180º which is not true for quads in. 2007, TESCCC Revised 10/01/07 page 5 of 50
6 Quadrilaterals and Angle Sums Practice (pg 2) KEY Practice Problems Use your knowledge of Triangle and Quadrilateral angle sums to complete the table below. Given Find m EAB=95 o m ABC=104 o m 8= 18º m 9= 115º m BED=80 o AC CD m 10= 60º m 11= 82º m 1=83 o m 2=62 o m 12= 100º m 13= 97º m 3=101 o m 4=104 o m 14= 93º m 15= 75º m 5=46 o m 6=70 o m 16= 35º m 17= 17º m 7=35 o m 18= 32º m 19= 14º B 16 A C 1 E G D F 2007, TESCCC Revised 10/01/07 page 6 of 50
7 Quadrilaterals and Angle Sums Quadrilaterals are four (quad) sided (lateral) geometric figures. Quadrilaterals can be convex or concave. CONVEX CONCAVE Investigation: Angles in a convex quadrilateral 1. Draw a foursided convex quadrilateral on a sheet of colored paper. 2. Measure the four angles to the nearest degree. Record your measures below. 3. Find the sum of the measure of the four angles. 4. Tear off the four angles and arrange them around the point below. 5. What conjecture can be made about the sum of the measures of the four angles of a convex quadrilateral? 6. Would this conjecture hold true for the sum of the angles of a concave quadrilateral? 2007, TESCCC Revised 10/01/07 page 7 of 50
8 Quadrilaterals and Angle Sums Practice (pg 2) Practice Problems Use your knowledge of Triangle and Quadrilateral angle sums to complete the table below. Given Find m EAB=95 o m ABC=104 o m 8= m 9= m BED=80 o AC CD m 10= m 11= m 1=83 o m 2=62 o m 12= m 13= m 3=101 o m 4=104 o m 14= m 15= m 5=46 o m 6=70 o m 16= m 17= m 7=35 o m 18= m 19= B 16 A C 1 E G D F 2007, TESCCC Revised 10/01/07 page 8 of 50
9 All in the Family KEY 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 parallel, one pair of opposite sides parallel, and neither pair of opposite sides 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 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. A Cand B D. Opposite angles appear to be congruent. b. Use your paper tracing to compare the side lengths of the parallelogram. What appears to be true? Summarize your findings below. AB=DC and AD=BC. Opposite sides appear to be congruent. 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. See student measures Consecutive angle pairs in a parallelogram are supplementary. d. Use a straightedge to draw the diagonals on the parallelogram above. 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? The diagonals of a parallelogram bisect each other. 2007, TESCCC Revised 10/01/07 page 9 of 50
10 All in the Family (pg 2) KEY 2. The figure below is a rectangle. A rectangle is a type of quadrilateral that has four right angles. H I K J suur suur suur suur a. Use you pencil and a straightedge to draw lines HI, KJ, and HK. Let HK be a transversal for the other two. What conclusions can you make about lines suur suur HI and KJ based on your knowledge of lines and transversals? Explain. See student drawings suur suur KHI and HKJ are supplementary; therefore, HI KJ. b. Use your pencil to draw sur suur suur sur IJ. Let KJ be a transversal for HK and IJ. What conclusions suur sur can you make about lines HK and IJ based on your knowledge of lines and transversals? Explain. suur sur HKJ and KJI are supplementary; therefore, HK IJ. c. What does your findings from part a and b lead you to believe about the rectangle? The rectangle is also a parallelogram. 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? See student drawing The diagonals appear to be congruent. 2007, TESCCC Revised 10/01/07 page 10 of 50
11 All in the Family (pg 3) KEY 3. The figure below is a rhombus. A rhombus is a type of quadrilateral that has four congruent sides. P S Q a. Find the measure of the angles of the rhombus. Record your answers below. See student measures R b. Let SR sur be a transversal for SP sur and suur RQ. What conclusions can you make about lines sur suur SP and RQ based on your knowledge of lines and transversals? Explain. sur suur PSR and SRQ are supplementary; therefore, SP RQ. c. Let SP sur be a transversal for SR sur and suur PQ. What conclusions can you make about lines sur suur SR and PQ based on your knowledge of lines and transversals? Explain. sur suur RSP and SPQ are supplementary; therefore, SR PQ. d. What do you conclusions in parts b and c lead you to believe about the rhombus? Explain. The rhombus is also a parallelogram. 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. See student measures The diagonals of a rhombus are perpendicular. 2007, TESCCC Revised 10/01/07 page 11 of 50
12 All in the Family (pg 4) KEY 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. Since the square has four right angles it is also a rectangle. Since the square has four congruent sides it is also a rhombus. Since the rectangle and rhombus are both types of parallelograms, the square is also a parallelogram. 5. Based on the previous explorations, what similarities did you discover about the parallelogram, rectangle, rhombus, and square? Similarities All are parallelograms Rectangle and square have 4 right angles Square and rhombus have 4 congruent sides Rectangle and square have congruent diagonals Square and rhombus have perpendicular diagonals 2007, TESCCC Revised 10/01/07 page 12 of 50
13 All in the Family (pg 5) KEY 6. The figure below is a trapezoid. A trapezoid is a type of quadrilateral with exactly one pair of parallel sides. A 1 2 B D 4 3 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. See student measures In the figure above, 1 and 4, and 2 and 3 are supplementary pairs; therefore, suur suur AB DC. 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. See student samples c. MN is called the median for the trapezoid. What do you think is true about points M and N? M and N are the midpoints of sides suur AD and suur BC. (students may label either order ) 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? See student samples It is equal to MN. e. Based on your answer to part d, what conjecture can your write about the median of a trapezoid? The median of a trapezoid has length equal to the average of the lengths of the bases of the trapezoid. 2007, TESCCC Revised 10/01/07 page 13 of 50
14 All in the Family (pg 6) KEY 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. See student samples b. Based on your folding, what can you conclude about the parts of the trapezoid? T Rand P A. c. Based on you findings in part a and b, write a conjecture about isosceles trapezoids. In an isosceles trapezoid, pairs of base angles are congruent. 2007, TESCCC Revised 10/01/07 page 14 of 50
15 All in the Family (pg 7) KEY 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 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 S 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. See student samples 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? The diagonals of a kite are perpendicular. d. Fold and crease your patty paper tracing so that E coincides sur with I. Examine the crease in your paper. What does this verify about diagonal EI? KT bisects EI. 2007, TESCCC Revised 10/01/07 page 15 of 50
16 All in the Family 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 parallel, one pair of opposite sides parallel, and neither pair of opposite sides 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 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 above. 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? 2007, TESCCC Revised 10/01/07 page 16 of 50
17 All in the Family (pg 2) 2. The figure below is a rectangle. A rectangle is a type of quadrilateral that has four right angles. H I K J suur suur suur suur a. Use you pencil and a straightedge to draw lines HI, KJ, and HK. Let HK be a transversal for the other two. What conclusions can you make about lines suur suur HI and KJ based on your knowledge of lines and transversals? Explain. b. Use your pencil to draw sur suur suur sur IJ. Let KJ be a transversal for HK and IJ. What conclusions suur sur can you make about lines HK and IJ based on your knowledge of lines and transversals? Explain. c. What does 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? 2007, TESCCC Revised 10/01/07 page 17 of 50
18 All in the Family (pg 3) 3. The figure below is a rhombus. A rhombus is a type of quadrilateral that has four congruent sides. P S Q a. Find the measure of the angles of the rhombus. Record your answers below. R b. Let SR sur be a transversal for SP sur and suur RQ. What conclusions can you make about lines sur suur SP and RQ based on your knowledge of lines and transversals? Explain. c. Let SP sur be a transversal for SR sur and suur PQ. What conclusions can you make about lines sur suur SR and PQ based on your knowledge of lines and transversals? Explain. d. What do you 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. 2007, TESCCC Revised 10/01/07 page 18 of 50
19 All in the Family (pg 4) 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? 2007, TESCCC Revised 10/01/07 page 19 of 50
20 All in the Family (pg 5) 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 do you think 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 your write about the median of a trapezoid? 2007, TESCCC Revised 10/01/07 page 20 of 50
21 All in the Family (pg 6) 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 you findings in part a and b, write a conjecture about isosceles trapezoids. 2007, TESCCC Revised 10/01/07 page 21 of 50
22 All in the Family (pg 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 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 S 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 sur with I. Examine the crease in your paper. What does this verify about diagonal EI? 2007, TESCCC Revised 10/01/07 page 22 of 50
23 Properties of Quadrilaterals (KEY) Properties of convex quadrilaterals: Have four sides. Have four vertices and angles. Sum of the angles equals 360 o. Are congruent if their corresponding angles and corresponding sides are congruent. Quadrilaterals are generally classified by the number of parallel sides they contain. Study the definitions below. Trapezium a quadrilateral with no pairs of parallel sides o Kite two congruent pairs of adjacent sides Property Diagonals are perpendicular. One of the diagonals bisects the other. Trapezoid a quadrilateral that has only one pair of parallel sides o Isosceles trapezoid non parallel legs are congruent Property The base angles of an isosceles trapezoid are congruent. Parallelogram a quadrilateral with two pair of parallel sides, opposite sides are parallel Properties Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent. Consecutive angles of a parallelogram are supplementary. The diagonals of a parallelogram bisect each other. o Rectangle parallelogram with four right angles Property The diagonals of a rectangle are congruent. o Rhombus parallelogram with four congruent sides Property The diagonals of a rhombus are perpendicular to each other. o Square parallelogram with four right angles and four congruent sides Practice Problems 1. Write a proof for the statement, Consecutive angles of a parallelogram are supplementary. Extend lines through the opposite sides of a parallelogram. Let one of the other sides be a transversal for the opposite sides. Since the opposite sides are parallel, the pair of same side interior angles formed is supplementary. The same side interior angles formed is a consecutive pair for the parallelogram. 2007, TESCCC Revised 10/01/07 page 23 of 50
24 Properties of Quadrilaterals (pg 2) KEY 2. Write a proof for the statement, The diagonals of a rhombus are perpendicular. R B H Since a rhombus is a parallelogram, the diagonals bisect each other. Therefore, ΔMBR ΔHBR by SSS, and MBR HBR by CPCTC. Since MBR and HBR are congruent and form a linear pair, each has a measure of 90º; therefore, RO MH (the diagonals are perpendicular). M O 2007, TESCCC Revised 10/01/07 page 24 of 50
25 Properties of Quadrilaterals (pg 3) KEY 3. In the parallelogram below, PG = 2x 7, MR = x + 5, and MG = 2x 5. Find the value of x and the PG, MR, and MG. P G M R x = 12, PG = 17, MR = 17, MG = Use the information in the rectangle below to find the value of x, the value of y, TE, RC, RX, EX, TX, and CX. R x E y P 3 T 4 C x = 4, y = 3, TE = 5, RC = 5, RP = EP = TP = CP = , TESCCC Revised 10/01/07 page 25 of 50
26 Properties of Quadrilaterals (pg 4) KEY 5. Given the formula for the area of a triangle, A triangle = ½(base)(height), use the properties of quadrilaterals to derive the formula for the area of a rhombus in terms of its diagonals. Triangle 1 Triangle 2 d 2 d 1 Since the rhombus is a parallelogram, the diagonals bisect each other resulting in four congruent triangles by SSS. Since the diagonals are perpendicular, the triangles are right 1 triangles. The legs of each triangle are 1 2 d and 1 d 2 ; and, the area of one of the triangles is 2 1 ( 1 d1)( 1 d 1 2) = d1d2. Therefore, the area of all four triangles (the rhombus) is ( 1 2) 8 dd = dd. 2007, TESCCC Revised 10/01/07 page 26 of 50
27 Properties of Quadrilaterals (pg 5) KEY 6. Suppose a carpenter is framing a rectangular room and wants to verify that the 4sided room is square (meaning each corner forms a right angle). What might the carpenter do to verify that each corner forms a right angle without measuring the angles? Assuming opposite sides are congruent, the carpenter could measure the lengths of the diagonals and adjust the framing until each diagonal is the same length, since diagonals of a rectangle are congruent. 7. Suppose a room is constructed in the shape of a rhombus so that one diagonal is 6 ft. long and the other is 8 ft. long. Find the perimeter of the rhombus. The perimeter is 20 ft. 2007, TESCCC Revised 10/01/07 page 27 of 50
28 Properties of Quadrilaterals (pg 6) KEY 8. Use the information in the trapezoid below to find the value of x, the value of y, m T, m R,and m P. X= 5, y = 20, m T = 120, m P= 60, and m R= 100 T (12x+60)º (5y)º R P (4x+40)º 80º A 9. Suppose the length of EI is 10 ft. Use the information in the kite below to find the perimeter of the kite. Perimeter is ( ) 27.1ft. K 45º 45º E 60º I 30º T 10. Use the information in the trapezoid below to find HK, IJ, m KHI HK = 12, IJ= 12, m KHI = 120, m KJI = 60, m HIJ = 120 H 6 3 I, m KJI and m HIJ. K 60º J 2007, TESCCC Revised 10/01/07 page 28 of 50
29 Properties of Quadrilaterals Properties of convex quadrilaterals: Have four sides. Have four vertices and angles. Sum of the angles equals 360 o. Are congruent if their corresponding angles and corresponding sides are congruent. Quadrilaterals are generally classified by the number of parallel sides they contain. Study the definitions below. Trapezium a quadrilateral with no pairs of parallel sides o Kite two congruent pairs of adjacent sides Property Diagonals are perpendicular. One of the diagonals bisects the other. Trapezoid a quadrilateral that has only one pair of parallel sides o Isosceles trapezoid non parallel legs are congruent Property The base angles of an isosceles trapezoid are congruent. Parallelogram a quadrilateral with two pair of parallel sides, opposite sides are parallel Properties Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent. Consecutive angles of a parallelogram are supplementary. The diagonals of a parallelogram bisect each other. o Rectangle parallelogram with four right angles Property The diagonals of a rectangle are congruent. o Rhombus parallelogram with four congruent sides Property The diagonals of a rhombus are perpendicular to each other. o Square parallelogram with four right angles and four congruent sides Practice Problems 1. Write a proof for the statement, Consecutive angles of a parallelogram are supplementary. 2007, TESCCC Revised 10/01/07 page 29 of 50
30 Properties of Quadrilaterals (pg 2) 2. Write a proof for the statement, The diagonals of a rhombus are perpendicular. R H B M O 2007, TESCCC Revised 10/01/07 page 30 of 50
31 Properties of Quadrilaterals (pg 3) 3. In the parallelogram below, PG = 2x 7, MR = x + 5, and MG = 2x 5. Find the value of x and the PG, MR, and MG. P G M R 4. Use the information in the rectangle below to find the value of x, the value of y, TE, RC, RX, EX, TX, and CX. R x E y P 3 T 4 C 2007, TESCCC Revised 10/01/07 page 31 of 50
32 Properties of Quadrilaterals (pg 4) 5. Given the formula for the area of a triangle, A triangle = ½(base)(height), use the properties of quadrilaterals to derive the formula for the area of a rhombus in terms of its diagonals. Triangle 1 Triangle 2 d 2 d , TESCCC Revised 10/01/07 page 32 of 50
33 Properties of Quadrilaterals (pg 5) 6. Suppose a carpenter is framing a rectangular room and wants to verify that the 4sided room is square (meaning each corner forms a right angle). What might the carpenter do to verify that each corner forms a right angle without measuring the angles? 7. Suppose a room is constructed in the shape of a rhombus so that one diagonal is 6 ft. long and the other is 8 ft. long. Find the perimeter of the rhombus. 2007, TESCCC Revised 10/01/07 page 33 of 50
34 Properties of Quadrilaterals (pg 6) 8. Use the information in the trapezoid below to find the value of x, the value of y, m T, m R,and m P. T (12x+60)º (5y)º R P (4x+40)º 80º A 9. Suppose the length of EI is 10 ft. Use the information in the kite below to find the perimeter of the kite. K 45º 45º E 60º I 30º T 10. Use the information in the trapezoid below to find HK, IJ, m KHI, m KJI and m HIJ. H I 6 3 K 60º J 2007, TESCCC Revised 10/01/07 page 34 of 50
35 Quadrilaterals and Coordinate (KEY) Part A Draw figure ABCD using the following ordered pairs: A(0, 0), B(3, 4), C(1, 7), and D(4, 3). Complete the table below. Length of the four sides: AB = 5 CD = 5 BC = 5 DA = 5 Length of the diagonals: AC = 5 2 BD = 5 2 Angle measures at each vertex: m DAB =90º m BCD =90º m ABC =90º m CDA =90º Length of the diagonal segments: AE = BE = EC = ED = Slope of the four sides: Slope of AB = 4/3 Slope of CD = 4/3 Slope of BC = 3/4 Slope of DA = 3/4 Slope of the diagonals: Slope of AC = 7/1 Slope of BD = 1/7 Point of intersection of the diagonals (Point E) (.5, 3.5) Angle measures of angles formed by diagonals: m AEB = 90º m CED = 90º m BEC = 90º m DEA = 90º How do you know that figure ABCD is a square? It has 4 right angles and 4 congruent sides. 2007, TESCCC Revised 10/01/07 page 35 of 50
36 Quadrilaterals and Coordinate (pg 2) KEY Squares 1. Write some conjectures you have about properties of squares and how the data you collected supports those conjectures. Opposite reciprocal slopes of sides verify right angles of the square. Opposite reciprocal slopes of diagonals verify that the diagonals are perpendicular. Length of diagonal segments are half the length of the diagonals, therefore, the diagonals bisect each other. 2. Create a square on the coordinate grid below that satisfies the following two conditions: a. The origin is not a vertex. b. No side is parallel to a coordinate axis. See student samples 3. How do you know that this figure is a square? It has 4 right angles and 4 congruent sides. 2007, TESCCC Revised 10/01/07 page 36 of 50
37 Quadrilaterals and Coordinate (pg 3) KEY Part B Draw figure ABCD using the following ordered pairs: A(0, 0), B(6, 8), C(2, 11), and D(4, 3). Complete the table below. Length of the four sides: AB =10 CD =10 BC = 5 DA = 5 Length of the diagonals: AC =5 5 BD = 5 5 Angle measures at each vertex: m DAB =90º m BCD =90º m ABC =90º m CDA =90º Length of the diagonal segments: AE = BE = EC = ED = Slope of the four sides: Slope of AB = 4/3 Slope of CD = 4/3 Slope of BC = 3/4 Slope of DA = 3/4 Slope of the diagonals: Slope of AC = 11/2 Slope of BD = 5/10 Point of intersection of the diagonals (Point E) (1, 5.5) Angle measures of angles formed by diagonals: (approx.) m AEB 127º m CED 127º m BEC 53º m DEA 53º How do you know that figure ABCD is a rectangle and not a square? It has 4 right angles but does not have 4 congruent sides. 2007, TESCCC Revised 10/01/07 page 37 of 50
38 Quadrilaterals and Coordinate (pg 4) KEY Rectangles 4. Write some conjectures you have about properties of rectangles and how the data you collected supports those conjectures. Opposite reciprocal slopes of sides verify right angles of the rectangle. The data shows that the lengths of the diagonals of a rectangle are equal. Length of diagonal segments are half the length of the diagonals, therefore, the diagonals bisect each other. 5. Create a rectangle on the coordinate grid below that satisfies the following three conditions: a. The origin is not a vertex. b. No side is parallel to a coordinate axis. c. The figure is not a square. See student samples 6. How do you know that this figure is a rectangle? The figure has 4 right angles. 2007, TESCCC Revised 10/01/07 page 38 of 50
39 Quadrilaterals and Coordinate (pg 5) KEY Part C Draw figure ABCD using the following ordered pairs: A(0, 0), B(5, 5), C(6, 12), and D(1, 7). Complete the table below. Length of the four sides: AB = 5 2 CD = 5 2 BC = 5 2 DA = 5 2 Length of the diagonals: AC = 6 5 BD = 2 5 Angle measures at each vertex approx. m DAB 36.9 m BCD 36.9 m ABC m CDA Length of the diagonal segments: AE =3 5 BE = 5 EC =3 5 ED = 5 Slope of the four sides: Slope of AB = 1/1 Slope of CD = 1/1 Slope of BC = 7/1 Slope of DA = 7/1 Slope of the diagonals: Slope of AC = 1/2 Slope of BD = 1/2 Point of intersection of the diagonals (Point E) (3, 6) Angle measures of angles formed by diagonals: m AEB = 90 m CED = 90 m BEC = 90 m DEA = 90 How do you know that figure ABCD is a rhombus and not a square? It has 4 congruent sides but does not have four right angles. 2007, TESCCC Revised 10/01/07 page 39 of 50
40 Quadrilaterals and Coordinate (pg 6) KEY Rhombi 7. Write some conjectures you have about properties of rhombi and how the data you collected supports those conjectures. The slopes of the diagonals are opposite reciprocals verifies that the diagonals are perpendicular. The lengths of the diagonal segments are half the length of the diagonals verifies that the diagonals bisect each other. 8. Create a rhombus on the coordinate grid below that satisfies the following three conditions: a. The origin is not a vertex. b. No side is parallel to a coordinate axis. c. The figure is not a square. See student samples 9. How do you know that this figure is a rhombus? The figure has four congruent sides. 2007, TESCCC Revised 10/01/07 page 40 of 50
41 Quadrilaterals and Coordinate Part A Draw figure ABCD using the following ordered pairs: A(0, 0), B(3, 4), C(1, 7), and D(4, 3). Complete the table below. Length of the four sides: AB = CD = BC = DA = Length of the diagonals: AC = BD = Angle measures at each vertex: m DAB = m BCD = m ABC = m CDA = Length of the diagonal segments: AE = BE = EC = ED = Slope of the four sides: Slope of AB = Slope of CD = Slope of BC = Slope of DA = Slope of the diagonals: Slope of AC = Slope of BD = Point of intersection of the diagonals (Point E) Angle measures of angles formed by diagonals: m AEB = m CED = m BEC = m DEA = How do you know that figure ABCD is a square? 2007, TESCCC Revised 10/01/07 page 41 of 50
42 Quadrilaterals and Coordinate (pg 2) Squares 1. Write some conjectures you have about properties of squares and how the data you collected supports those conjectures. 2. Create a square on the coordinate grid below that satisfies the following two conditions: a. The origin is not a vertex. b. No side is parallel to a coordinate axis. 3. How do you know that this figure is a square? 2007, TESCCC Revised 10/01/07 page 42 of 50
43 Quadrilaterals and Coordinate (pg 3) Part B Draw figure ABCD using the following ordered pairs: A(0, 0), B(6, 8), C(2, 11), and D(4, 3). Complete the table below. Length of the four sides: AB = CD = BC = DA = Length of the diagonals: AC = BD = Angle measures at each vertex: m DAB = m BCD = m ABC = m CDA = Length of the diagonal segments: AE = BE = EC = ED = Slope of the four sides: Slope of AB = Slope of CD = Slope of BC = Slope of DA = Slope of the diagonals: Slope of AC = Slope of BD = Point of intersection of the diagonals (Point E) Angle measures of angles formed by diagonals: m AEB m CED m BEC m DEA How do you know that figure ABCD is a rectangle and not a square? 2007, TESCCC Revised 10/01/07 page 43 of 50
44 Quadrilaterals and Coordinate (pg 4) Rectangles 4. Write some conjectures you have about properties of rectangles and how the data you collected supports those conjectures. 5. Create a rectangle on the coordinate grid below that satisfies the following three conditions: c. The origin is not a vertex. d. No side is parallel to a coordinate axis. e. The figure is not a square. 6. How do you know that this figure is a rectangle? 2007, TESCCC Revised 10/01/07 page 44 of 50
45 Quadrilaterals and Coordinate (pg 5) Part C Draw figure ABCD using the following ordered pairs: A(0, 0), B(5, 5), C(6, 12), and D(1, 7). Complete the table below. Length of the four sides: AB = CD = BC = DA = Length of the diagonals: AC = BD = Angle measures at each vertex m DAB m BCD m ABC m CDA Length of the diagonal segments: AE = BE = EC = ED = Slope of the four sides: Slope of AB = Slope of CD = Slope of BC = Slope of DA = Slope of the diagonals: Slope of AC = Slope of BD = Point of intersection of the diagonals (Point E) Angle measures of angles formed by diagonals: m AEB = m CED = m BEC = m DEA = How do you know that figure ABCD is a rhombus and not a square? 2007, TESCCC Revised 10/01/07 page 45 of 50
46 Quadrilaterals and Coordinate (pg 6) Rhombi 7. Write some conjectures you have about properties of rhombi and how the data you collected supports those conjectures. 8. Create a rhombus on the coordinate grid below that satisfies the following three conditions: a. The origin is not a vertex. b. No side is parallel to a coordinate axis. c. The figure is not a square. 9. How do you know that this figure is a rhombus? 2007, TESCCC Revised 10/01/07 page 46 of 50
47 Trace the Ancestry KEY 1. Create a Family Tree of quadrilaterals based on their properties and parallel sides that shows how the various quadrilaterals are related. See diagram below Quadrilaterals Trapeziums Trapezoids Parallelograms Kites Isosceles Trapezoid Rectangle Rhombus Square 2007, TESCCC Revised 10/01/07 page 47 of 50
48 Trace the Ancestry (pg 2) KEY 2. For the quadrilateral represented by the ordered pairs A(0, 4), B(4, 0), C(0, 4), D(4, 0): a. Graph the figure, labeling points. See student samples b. Name the quadrilateral. Square c. Find the length of each side. 4 2 d. Find the slope of each side. Slope AD is 1, Slope CD is 1, Slope BC is 1, Slope AB is 1. e. Identify all types of quadrilaterals it represents. Square, Rectangle, Rhombus, Parallelogram f. Justify your conclusions. Square Opposite reciprocal slopes (1 and 1) of sides verify right angles of the square; all side lengths are 4 2. RectangleThe Square has 4 right angles. RhombusThe Square has 4 congruent sides. ParallelogramThe Square has opposite sides parallel since consecutive angles are supplementary. 2007, TESCCC Revised 10/01/07 page 48 of 50
49 Trace the Ancestry 1. Create a Family Tree of quadrilaterals based on their properties and parallel sides that shows how the various quadrilaterals are related. 2007, TESCCC Revised 10/01/07 page 49 of 50
50 Trace the Ancestry (pg 2) 2. For the quadrilateral represented by the ordered pairs A(0, 4), B(4, 0), C(0, 4), D(4, 0): a. Graph the figure, labeling points. b. Name the quadrilateral. c. Find the length of each side. d. Find the slope of each side. e. Identify all types of quadrilaterals it represents. f. Justify your conclusions. 2007, TESCCC Revised 10/01/07 page 50 of 50
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