Quadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid


 Phillip Waters
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1 Quadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid Grade level: 10 Prerequisite knowledge: Students have studied triangle congruences, perpendicular lines, parallel lines, angle congruences such as corresponding angles and alternate interior angles, and related proofs Textbook used: Integrated Mathematics Course 11 (Second Edition) Authors: Edward P. Keenan and Isidore Dressler Publisher: Amsco School Publications, Inc. Publication date: 1990 Workbook used: Geometry Publisher: Instructional Fair, Inc. Publication date: 1994 ISBN
2 Quadrilaterals Day 1: Day 2: Day 3: Day 4: Day 5: Definition of a quadrilateral Terms referring to the parts of a quadrilateral Definition of a parallelogram Properties of a parallelogram Ways to prove that a quadrilateral is a parallelogram Proofs Definition of a rectangle Definition of a rhombus Definition of a square Properties of a rectangle Properties of a rhombus Properties of a square Definition of a trapezoid Definition of an isosceles trapezoid Properties of a trapezoid Properties of an isosceles trapezoid Activity on quadrilaterals and their properties Summary of quadrilaterals and their properties 2
3 Overview Day 1: Day 2: Day 3: Day 4: Day 5: The lesson begins with a discussion in regards to the definition of a quadrilateral and the terms referring to the parts of the quadrilateral. Students work in pairs to construct a parallelogram and discover properties of a parallelogram. This is done on a computer using Geometer s Sketchpad software. Students will summarize these properties and a worksheet is given to reinforce the lesson. The lesson begins with a review of the properties of a parallelogram. Students think of ways to prove that a quadrilateral is a parallelogram. Students are given two worksheets to complete on proving a quadrilateral is a parallelogram. The teacher guides students through these proofs and theorems. Students discuss the definitions of a rectangle, a rhombus, and a square. Then students work in pairs to construct these quadrilaterals and discover their properties with the aid of Geometer s Sketchpad software on a computer. The class as a group summarize these properties. Another worksheet is given to reinforce this lesson. Students discuss the definitions of a trapezoid and an isosceles trapezoid. Students work in pairs to construct these quadrilaterals and discover their properties with the aid of Geometer s Sketchpad software on a computer. Then the class as a group summarize these properties. Another worksheet is given to reinforce the lesson. To reinforce this unit, students are guided through a group activity in which they use a rope to create quadrilaterals. Then students complete a review worksheet on the properties of these quadrilaterals. 3
4 Student Objectives Day 1: Day 2: Day 3: Day 4: Day 5: Students will understand what a quadrilateral is and also have an understanding of what a parallelogram is. Students will understand ways to prove that a quadrilateral is a parallelogram. Students will also understand that a diagonal divides a parallelogram into two congruent triangles. Students will understand the properties of a rectangle, a rhombus, and a square. Students will understand the properties of a trapezoid and an isosceles trapezoid. Students can identify different quadrilaterals and identify their properties. An activity is done to reinforce the lessons and to strengthen the students' spatial sense. 4
5 Performance Standards NYS Core Curriculum Performance Indicators Operation Using computers to analyze mathematical phenomena (My lessons require computers and Geometer s Sketchpad software to discover properties of some quadrilaterals.) Modeling/Multiple Representation Use learning technologies to make and verify geometric conjectures Justify the procedures for basic geometric constructions (My lessons require computers and Geometer s Sketchpad software to construct some quadrilaterals and to make conjectures regarding their properties.) Illustrate spatial relationships using perspective (The group activity that I chose as a review and reinforcement of the quadrilaterals and their properties involved spatial relationships as students formed different quadrilaterals using themselves as vertices and a rope to outline it.) Measurement Use geometric relationships in relevant measurement problems involving geometric concepts (Students used Geometer s Sketchpad software and measured lengths and angles. They related these measurements to their knowledge of angle congruences, such as alternate interior angles and corresponding angles.) Mathematical Reasoning Use geometric relationships in relevant measurement problems involving geometric concepts (Investigating angle measurements required students to use their knowledge of angle relationships and congruences.) Patterns/Functions Use computers to analyze mathematical phenomena (Students used computers with the aid of Geometer s Sketchpad to study the measure of lengths and angles of different quadrilaterals.) 5
6 Performance Standards NCTM Principles and Standards for School Mathematics Standard 1: Mathematics as Problem Solving (Students worked on computers using Geometer s Sketchpad software to discover some properties of different quadrilaterals through the measurements of segments and angles.) Standard 2: Mathematics as Communication (Students worked in pairs to discover properties of some quadrilaterals. In order to do so, they needed to communicate with each other. Students also needed to communicate as a group during the review activity on day 5. Also, students needed to communicate during the discussions leading to and summarizing the properties of these quadrilaterals.) Standard 3: Mathematics as Reasoning (Students made conjectures about some of the properties of the quadrilaterals an tested them using Geometer s Sketchpad by dragging the vertices to see if these properties still held.) 6
7 Equipment and environment An overhead projector and transparencies are used to present this unit. Students are given worksheets as a guide to constructions and investigations of different quadrilaterals using Geometer s Sketchpad software on computers. These constructions and investigations can be done in pairs. Additional worksheets to be done individually are also given to reinforce the lessons. On the final day, a group activity is done with the students to review properties of different quadrilaterals that were presented in this unit. A rope is used (as a manipulative) to do this activity. (I adapted this activity to my unit on quadrilaterals from a lesson a fellow teacher used in her class when presenting right triangles.) 7
8 Quadrilaterals Lesson Plan  Day 1 Student Objectives: Students will understand what a quadrilateral is and also have an understanding of what a parallelogram is. Equipment and Environment: An overhead projector and transparencies will be used. Students will work with a partner using computers and Geometer s Sketchpad software. Students will also be given worksheets. Opening Activity: Students will discuss quadrilaterals. Terms referring to the parts of a quadrilateral (vertices, consecutive vertices, consecutive sides, opposite sides, consecutive angles, opposite angles, and diagonals) will be discussed. A transparency will be displayed with these definitions. Students can also be given this information on a handout. Developmental Activity: Students will work in pairs to investigate the parallelogram. They will begin with the following definition of a parallelogram: A parallelogram is a quadrilateral with two pairs of opposite sides parallel. (This will also be shown on a transparency.) Students will each receive a worksheet entitled "Parallelogram: Construction and Investigation" which will guide as they construct a parallelogram and then measure its segments and angles. Students should be able to "discover" some properties of a parallelogram. Closing Activity: Students will summarize the properties of a parallelogram. These properties can then be displayed using a transparency on an overhead projector. As a final reinforcement, students can complete the worksheet on quadrilaterals entitled Properties of a Parallelogram which is page 63 from the Geometry workbook for homework. 8
9 Quadrilaterals: Related Terms The following will be presented to the class during the discussion on what a quadrilateral is. This will be shown as a transparency on an overhead projector. THE GENERAL QUADRILATERAL A quadrilateral, ABCD, is a polygon with four sides. A vertex is an endpoint where two sides meet. (A, B, C, or D) Consecutive vertices are vertices that are endpoints of the same side. (A and B, B and C, C and D, or D and A) Consecutive sides or Adjacent sides are those sides that have a common endpoint. (segments AB and BC, segments BC and CD, segments CD and DA, segments DA and AB) Opposite sides of a quadrilateral are sides that do not have a common endpoint. (segments BC and DA, segments ) Consecutive angles are angles whose vertices are consecutive. (angles DAB and ABC, angles ABC and BCD, angles BCD and CDA, angles CDA and DAB) Opposite angles are angles whose vertices are not consecutive. (angles DAB and BCD, angles) A diagonal is a line segment that joins two vertices that are not consecutive. (segment AC and segment BD) 9
10 Quadrilaterals Parallelogram: Construction and Investigation Using Geometer's Sketchpad, construct a parallelogram. 1. Construct a line. 2. Construct a point not on the line. 3. Select the line and the point. 4. Construct a line parallel to the first line, through that point. 5. Select one of the lines. Construct a point on the line (construct point on object). 6. Select the other line. Construct a point on the line (construct point on object). 7. Select those points and construct segment. 8. Select one of the lines. Construct a point on the line (construct point on object). 9. Select this point and the segment. 10. Construct a parallel line. 11. Construct point at intersection. 12. Select lines. Hide lines. 13. Select points. Construct segments. 14. Label vertices A, B, C, and D. 15. Select segments. Measure segments. 16. Measure angles. 17. Drag the vertices. What happens to the lengths and angle measures? 18. Construct the diagonals. 19. Select the diagonals. Construct point at intersection. Label this point, E. 20. Select the diagonals. Measure their lengths. 21. Select the diagonals. Hide those segments. 22. Construct segments from vertices to point E. Measure the segments. 23. Measure the angles. 24. Drag the vertices. What happens to the lengths and angle measures? 10
11 Answers to Parallelogram: Construction and Investigation Using Geometer's Sketchpad, construct a parallelogram. 1. Construct a line. 2. Construct a point not on the line. 3. Select the line and the point. 4. Construct a line parallel to the first line, through that point. 5. Select one of the lines. Construct a point on the line (construct point on object). 6. Select the other line. Construct a point on the line (construct point on object). 7. Select those points and construct segment. 8. Select one of the lines. Construct a point on the line (construct point on object). 9. Select this point and the segment. 10. Construct a parallel line. 11. Construct point at intersection. 12. Select lines. Hide lines. 13. Select points. Construct segments. 14. Label vertices A, B, C, and D. 15. Select segments. Measure segments. 16. Measure angles. Opposite sides and opposite angles are congruent. Consecutive angles are supplementary. 17. Drag the vertices. What happens to the lengths and angle measures? Opposite sides and opposite angles are still congruent. Consecutive angles are still supplementary. 18. Construct the diagonals. 19. Select the diagonals. Construct point at intersection. Label this point, E. 20. Select the diagonals. Measure their lengths. 21. Select the diagonals. Hide those segments. 22. Construct segments from vertices to point E. Measure the segments. 23. Measure the angles. The diagonals bisect each other. Alternate interior angles are congruent 24. Drag the vertices. What happens to the lengths and angle measures? Although the lengths and angle measures change, the diagonals still bisect each other, and the alternate interior angles are still congruent. 11
12 Quadrilaterals Properties of a Parallelogram 1. Opposite sides of a parallelogram are parallel. 2. Opposite sides of a parallelogram are congruent. 3. Opposite angles of a parallelogram are congruent. 4. Two consecutive angles of a parallelogram are supplementary. 5. Diagonals of a parallelogram bisect each other. 12
13 Quadrilaterals Lesson Plan  Day 2 Student Objectives: Students will understand ways to prove that a quadrilateral is a parallelogram. Students will also understand that a diagonal divides a parallelogram into two congruent triangles. Equipment and Environment: An overhead projector and transparencies will be used. Students will also receive worksheets on quadrilaterals. The first one is entitled "Two Column Proofs" and the second one is entitled "More Two Column Proofs". Students will work independently during this lesson. These are pages 64 and 65 from the Geometry workbook. Opening Activity: Students will review the properties of a parallelogram. The teacher will ask questions regarding the investigation students did during the previous lesson. The teacher can propose the following question: "How can we prove that a quadrilateral is a parallelogram?" Students should answer that if a quadrilateral has a property of a parallelogram it can be proven that it is a parallelogram. Discuss the ways to prove that a quadrilateral is a parallelogram. These can be displayed on a transparency on the overhead projector. Developmental Activity: Students will be given two worksheets on ways to prove that a quadrilateral is a parallelogram. They will be given some time to work on these independently, as the teacher circulates to offer assistance, and then the proofs will be discussed. Closing Activity: Summarize the conditions that are sufficient to show that a quadrilateral is a parallelogram. These can be stated in theorem form and displayed using a transparency on the overhead projector. 13
14 Quadrilaterals Lesson Plan  Day 2 To prove that a quadrilateral is a parallelogram, prove that any one of the following statements is true: 1. Both pairs of opposite sides are parallel. 2. Both pairs of opposite sides are congruent. 3. One pair of opposite sides are congruent and parallel. 4. Both pairs of opposite angles are congruent. 5. The diagonals bisect each other. 14
15 Quadrilaterals Lesson Plan  Day 3 Student Objectives: Students will understand the properties of a rectangle, a rhombus, and a square. Equipment and Environment: An overhead projector and transparencies will be used. Students will work with a partner using computers Geometer's Sketchpad software. Students will also be given worksheets. Opening Activity: The definitions of a rectangle, a rhombus, and a square will be discussed. (A rectangle is a parallelogram, one of whose angles is a right angle. A rhombus is a parallelogram that has two consecutive sides. A square is a rectangle that has two congruent consecutive sides.) These definitions will also be show a transparency. Developmental Activity: Students will work in pairs to investigate properties of a rectangle, a rhombus, and a square. Students will each receive worksheets entitled "Rectangle: Construction and Investigation", "Rhombus: Construction and Investigation", and "Square: Construction and Investigation", respectively. These worksheets will guide them as they construct these quadrilaterals and measure their segments and angles. Students should be able to "discover" some properties of a rectangle, a rhombus, and a square. Closing Activity: Students need to summarize the properties of a rectangle, a rhombus, and a square. These properties can then be displayed using a transparency on an overhead projector. As a final reinforcement, students can complete the worksheet on quadrilaterals entitled "Special Parallelograms". This is page 66 from the Geometry workbook. 15
16 Quadrilaterals Rectangle: Construction and Investigation Using Geometer's Sketchpad, construct a rectangle. 1. Construct a line. 2. Construct point on object. 3. Select the line and point. Construct a perpendicular line. 4. Construct a point on object; that is, on this perpendicular line. 5. Select this point and line. Construct another perpendicular line. 6. Construct a point on object; that is, construct a point on that perpendicular line, to the left of the intersection. 7. Select the line and point. Construct a perpendicular line. 8. Construct a point at intersection, that is, the intersection of the last perpendicular line and the original line that was constructed. 9. Label points at intersection A, B, C, D. 10. Select lines. Hide lines. 11. Construct segments. 12. Select segments. Measure segments. 13. Measure angles. 14. Drag the vertices. What happens to the lengths and angle measures? 15. Construct the diagonals. 16. Select the diagonals. Construct point at intersection. Label this point, E. 17. Select the diagonals. Measure their lengths. 18. Select the diagonals. Hide those segments. 19. Construct segments from the vertices to point E. Measure the segments. 20. Measure the angles. 21. Drag the vertices. What happens to the lengths and the angle measures?
17 Quadrilaterals Rhombus: Construction and Investigation Using Geometer's Sketchpad, construct a rhombus. 1. Construct two intersecting lines. 2. Construct point at intersection. Label this point, A. 3. Select one of the lines. Construct point on object. Label this point, B. 4. Select point, B, and the other line. 5. Construct a line parallel to that line and through point, B. 6. Construct a circle with center at B and radius segment BA. Label the intersection to the left of B, C. 7. Construct a line through C parallel to segment AB. 8. Label new point of intersection, D. 9. Hide lines. 10. Construct segments. 11. Measure segments. 12. Measure angles. 13. Drag the vertices. 14. Construct the diagonals. 15. Label the point of intersection, E. 16. Measure the lengths of the diagonals. 17. Hide the diagonals. 18. Construct segments from vertices to point E. 19. Measure those segments. 20. Measure the angles. 21. Drag the vertices. 17
18 Quadrilaterals Square: Construction and Investigation Using Geometer's Sketchpad, construct a square. 1. Construct a segment. 2. Rotate the segment 90 degrees. 3. Again, rotate this segment 90 degrees. 4. Construct a segment connecting the open end to form a square. 5. Label vertices A, B, C, D. 6. Measure segments. 7. Measure angles. 8. Drag the vertices. 9. Construct the diagonals. 10. Label point of intersection, E. 11. Measure the length of the diagonals. 12. Hide the diagonals. 13. Construct segments from vertices to point E. 14. Measure those segments. 15. Measure the angles. 16. Drag the vertices. 18
19 Answers to Rectangle: Construction and Investigation Using Geometer's Sketchpad, construct a rectangle. 1. Construct a line. 2. Construct point on object. 3. Select the line and point. Construct a perpendicular line. 4. Construct a point on object; that is, on this perpendicular line. 5. Select this point and line. Construct another perpendicular line. 6. Construct a point on object; that is, construct a point on that perpendicular line to the left of the intersection. 7. Select the line and point. Construct a perpendicular line. 8. Construct a point at intersection; that is the intersection of the last perpendicular line and the original line that was constructed. 9. Label points at intersection A, B, C, D. 10. Select lines. Hide lines. 11. Construct segments. 12. Select segments. Measure segments. 13. Measure angles. Opposite sides are equal in length. Each angle is ninety degrees. 14. Drag the vertices. What happens to the lengths and angle measures? Opposite sides are equal in length. Each angle is still equal to ninety degrees. 15. Construct the diagonals. 16. Select the diagonals. Construct point at intersection. Label this point, E. 17. Select the diagonals. Measure their lengths. 18. Select the diagonals. Hide those segments. 19. Construct segments from the vertices to point E. Measure the segments. 20. Measure the angles. Opposite sides are congruent, the diagonals are equal to each other, and the consecutive angles and the opposite angles are supplementary. Each angle is equal to ninety degrees. The diagonals bisect each other. 21. Drag the vertices. What happens to the lengths and the angle measures? Opposite sides are still congruent. The diagonals are still equal to each other. The consecutive angles and the opposite angles are still equal to each other. 19
20 Answers to Rhombus: Construction and Investigation Using Geometer's Sketchpad, construct a rhombus. 1. Construct two intersecting lines. 2. Construct point at intersection. Label this point, A. 3. Select one of the lines. Construct point on object. Label this point, B. 4. Select point, B, and the other line. 5. Construct a line parallel to that line and through point, B. 6. Construct a circle with center at B and radius segment BA. 7. Construct a line through C parallel to segment AB. 8. Label new point of intersection, D. 9. Hide lines. 10. Construct segments. 11. Measure segments. 12. Measure angles. All four sides are congruent. Opposite angles are congruent Any two consecutive angles are supplementary. 13. Drag the vertices. All four sides are still congruent. Opposite angles are still congruent. Any two consecutive angles are still supplementary. 14. Construct the diagonals. 15. Label the point of intersection, E. 16. Measure the lengths of the diagonals. 17. Hide the diagonals. 18. Construct segments from vertices to point E. 19. Measure those segments. 20. Measure the angles. The diagonals bisect each other. The diagonals bisect its angles. The diagonals are perpendicular to each other. 21. Drag the vertices. The diagonals still bisect each other. The diagonals still bisect its angles. The diagonals are still perpendicular to each other. 20
21 Answers to Square: Construction and Investigation Using Geometer s Sketchpad, construct a square. 1. Construct a segment. 2. Rotate the segment 90 degrees. 3. Again, rotate this segment 90 degrees. 4. Construct a segment connecting the open end to form a square. 5. Label vertices A, B, C, D. 6. Measure segments. 7. Measure angles. All four sides are congruent. All four angles are congruent. Each angle is 90 degrees. 8. Drag the vertices. All four sides are still congruent. All four angles are still congruent. Each angle is still 90 degrees. 9. Construct the diagonals. 10. Label point of intersection, E. 11. Measure the length of the diagonals. 12. Hide the diagonals. 13. Construct segments from vertices to point E. 14. Measure those segments. 15. Measure the angles. The diagonals bisect each other. The diagonals bisect the angles of the square. The diagonals are perpendicular. The diagonals are congruent. 16. Drag the vertices. The diagonals still bisect each other. The diagonals still bisect the angles of the square. The diagonals are still perpendicular. The diagonals are still congruent. 21
22 Quadrilaterals Lesson Plan  Day 3 Properties of a Rectangle 1. A rectangle has all the properties of a parallelogram. 2. A rectangle has four right angles and is therefore equiangular. 3. The diagonals of a rectangle are congruent. Properties of a Rhombus 1. A rhombus has all the properties of a parallelogram. 2. A rhombus has four congruent sides and is therefore equilateral. 3. The diagonals of a rhombus are perpendicular to each other. 4. The diagonals of a rhombus bisect its angles. Properties of a Square 1. A square has all the properties of a rectangle. 2. A square has all the properties of a rhombus.
23 Quadrilaterals Lesson Plan  Day 4 Student Objectives: Students will understand the properties of a trapezoid and an isosceles trapezoid. Equipment and Environment: An overhead projector and transparencies will be used. Students will work with a partner using computers and Geometer's Sketchpad software. Students will also be given worksheets. Opening Activity: The definitions of a trapezoid and an isosceles trapezoid will be discussed. They will also be shown on a transparency. Developmental Activity: Students will work in pairs to investigate properties of a trapezoid and an isosceles trapezoid. Students will each receive worksheets entitled "Trapezoid: Construction and Investigation" and "Isosceles Trapezoid: Construction and Investigation", respectively. These worksheets will guide them as they construct these quadrilaterals and measure their segments and angles. Students should be able to "discover" some properties of a trapezoid and an isosceles trapezoid. Closing Activity: Students will summarize the properties of a trapezoid and an isosceles trapezoid. These properties can then be displayed using a transparency on the overhead projector. Finally, students can complete for homework, the worksheet on quadrilaterals entitled "Trapezoids" which is page 67 from the Geometry workbook. 23
24 Trapezoid: Construction and Investigation Using Geometer's Sketchpad, construct a trapezoid. 1. Construct a line. 2. Construct a point not on the line. 3. Construct a line parallel to the first line through the point. 4. Construct a point on the first line. 5. Construct a segment connecting this point to the point on the second line. 6. Construct another point on each line. 7. Construct a segment connecting these points. 8. Hide lines. 9. Construct segments. 10. Measure segments. 11. Measure angles. 12. Drag the vertices. 13. Construct the diagonals. 14. Label point of intersection, E. 15. Measure the length of the diagonals. 16. Hide the diagonals. 17. Construct segments from vertices to point E. 18. Measure those segments. 19. Measure the angles. 20. Drag the vertices. 21. Hide the diagonals. 22. Construct point at midpoint on each of the legs. 23. Construct a segment. 24. Measure the segment (median). 25. Compare this to the lengths of the bases. 26. Measure the angles. 27. Drag the vertices.
25 Isosceles Trapezoid: Construction and Investigation Using Geometer's Sketchpad, construct a trapezoid. 1. Construct a line. 2. Construct a point not on the line. Label this point B. 3. Construct a line parallel to the first line through point, B. 4. Construct a point on the first line. Label this point, A. 5. Construct a segment connecting this point to the point on the second line. 6. Construct another segment congruent to that segment: Construct a point on the second line. Label that point, C. 7. Construct a circle with center C and radius segment AB. Label a point where the circle crosses the first line, D. 8. Hide lines. 9. Construct segments. 10. Measure segments. 11. Measure angles. 12. Drag the vertices. 13. Construct the diagonals. 14. Label point of intersection, E. 15. Measure the length of the diagonals. 16. Hide the diagonals. 17. Construct segments from vertices to point E. 18. Measure those segments. 19. Measure the angles. 20. Drag the vertices. 21. Hide the diagonals. 22. Construct point at midpoint on each of the legs. 23. Construct a segment. 24. Measure the segment (median). 25. Compare this to the lengths of the bases. 26. Measure the angles. 27. Drag the vertices. 25
26 Answers to Trapezoid: Construction and Investigation Using Geometer's Sketchpad, construct a trapezoid. 1. Construct a line. 2. Construct a point not on the line. 3. Construct a line parallel to the first line through this point. 4. Construct a point on the first line. 5. Construct a segment connecting this point to the point on the second line. 6. Construct another point on each line. 7. Construct a segment connecting these points. 8. Hide lines. 9. Construct segments. 10. Measure segments. 11. Measure angles. Adjacent angles are supplementary. 12. Drag the vertices. Adjacent angles are still supplementary. 13. Construct the diagonals. 14. Label point of intersection, E. 15. Measure the length of the diagonals. 16. Hide the diagonals. 17. Construct segments from vertices to point E. 18. Measure those segments. 19. Measure the angles. The diagonals are not congruent and they do not bisect each other. 20. Drag the vertices. The diagonals are still not congruent and still do not bisect each other. 21. Hide the diagonals. 22. Construct point at midpoint on each of the legs. 23. Construct a segment. 24. Measure the segment (median). 25. Compare this to the lengths of the bases. 26. Measure the angles. The median is parallel to the bases and equal to onehalf of their sum. 27. Drag the vertices. The median is still parallel to the bases. 26
27 Answers to Isosceles Trapezoid: Construction and Investigation Using Geometer s Sketchpad, construct a trapezoid. 1. Construct a line. 2. Construct a point not on the line. Label this point, B. 3. Construct a line parallel to the first line through point, B. 4. Construct a point on the first line. Label this point, A. 5. Construct a segment connecting this point to the point on the second line. 6. Construct another segment congruent to that segment: Construct a point on the second line. Label that point, C. 7. Construct a circle with center C and radius segment AB. Label point where the circle crosses the first line, D. 8. Hide lines. 9. Construct segments. 10. Measure segments. 11. Measure angles. The base angles are congruent. The legs are congruent. 12. Drag the vertices. The base angles are still congruent and the legs are still congruent. 13. Construct the diagonals. 14. Label point of intersection, E. 15. Measure the length of the diagonals. 16. Hide the diagonals. 17. Construct segments from vertices to point E. 18. Measure those segments. 19. Measure the angles. The diagonals are congruent. The shorter segments of each diagonal are congruent and the longer segments are congruent 20. Drag the vertices. The diagonals are still congruent. The shorter segments of each diagonal are still congruent and the longer segments are congruent. 21. Hide the diagonals. 22. Construct point at midpoint on each of the legs. 23. Construct a segment. 24. Measure the segment (median). 25. Compare this to the lengths of the bases. 26. Measure the angles. The median is parallel to the bases. 27. Drag the vertices. The median is parallel to the bases. 27
28 Quadrilaterals Lesson Plan  Day 4 Properties of a Trapezoid 1. A trapezoid has four sides. 2. A trapezoid has only one pair of parallel sides. 3. The median is parallel to the bases. 4. The median has a length equal to the average of the bases. Properties of an Isosceles Trapezoid 1. An isosceles trapezoid has all the properties of a trapezoid. 2. An isosceles trapezoid has congruent legs. 3. The base angles of an isosceles trapezoid are congruent. 4. The diagonals are congruent. 28
29 Quadrilaterals Lesson Plan  Day 5 Student Objectives: Students can identify different quadrilaterals and identify their properties. Equipment and Environment: An overhead projector and transparencies will be used. A rope will be used during the group activity. Each student will also receive a worksheet to be done independently. Opening Activity: The teacher begins by asking the students, "How many sides does a quadrilateral have?" Since the answer is four, the teacher then asks for four volunteers. This activity will reinforce different kinds of quadrilaterals and strengthen the students' knowledge of quadrilaterals and spatial sense. The four volunteers go to the front of the room and are asked to form a parallelogram by positioning themselves as the vertices. Then hand the students a rope to hold to outline the parallelogram they have made. Ask the students, "Why is this a parallelogram?" and have them discuss the properties of a parallelogram. Then choosing four new volunteers, have them go to the front of the room and form a rectangle by positioning themselves as the vertices. Hand them the rope to hold to outline the rectangle that they have made. Ask the students, "Why is this a rectangle?" and have them discuss the properties of a rectangle. Repeat this procedure for a rhombus, a square, a trapezoid, and an isosceles trapezoid. Developmental Activity: Each of the students will be given a worksheet entitled "A Summary of the Properties of Different Quadrilaterals". Students will be given some time to complete the worksheet. This should be done independently. Closing Activity: Display a completed chart using a transparency on an overhead projector and discuss the answers. 29
30 Quadrilaterals Lesson Plan  Day 5 A Summary of the Properties of Different Quadrilaterals Under the letters "a" through "g" in the following table, answer "yes" or "no" to the following questions for each of the given quadrilaterals. a. Are opposite sides congruent and parallel? b. Are opposite angles congruent? c. Are the diagonals congruent? d. Do the diagonals bisect each other? e. Are the diagonals perpendicular to each other? Are all angles congruent? g. Are any two consecutive sides congruent? a b c d e f g Parallelogram Rectangle Rhombus Square Trapezoid 30
31 Answers to Quadrilaterals Lesson Plan  Day 5 A Summary of the Properties of Different Quadrilaterals Under the letters "a" through "g" in the following table, answer "yes" or "no" to the following questions for each of the given quadrilaterals. a. Are opposite sides congruent and parallel? b. Are opposite angles always congruent? c. Are the diagonals always congruent? d. Do the diagonals bisect each other? e. Are the diagonals perpendicular to each other? f. Are all angles congruent? g. Are any two consecutive sides always congruent? a b c d e f g Parallelogram yes yes no yes no no no Rectangle yes yes yes yes no yes no Rhombus yes yes no yes yes no yes Square yes yes yes yes yes yes yes Trapezoid no no no no no no no
32 Quadrilaterals Assessments Students will be evaluated by their performance on the following test, the worksheets, and the homework assignments. These assessments include performance indicators from the NYS performance standards. One is measurement, that is, using geometric relationships in relevant measurement problems involving geometric concepts. Some of the problems on both the test and the worksheets involve finding the missing measurements. Also, mathematical reasoning, another performance standard, is met. Students need to use their previous knowledge of mathematical concepts and apply this knowledge when solving mathematical problems. And the worksheets address these NYS performance standards: Operation, Modeling/Multiple Representation, and Patterns/Functions because they require students to use computers or technology. 32
33 Test on Quadrilaterals 1. (6 points) In a parallelogram ABCD, if the measurement of angle B exceeds the measurement of angle A by 50, find the degree measure of angle B. 2. Given: ABCD is a parallelogram. E is the midpoint of segment AB. F is the midpoint of segment DC. Prove: EBFD is a parallogram. (10 points) 3. In rectangle ABCD, CB = 6, AB = 8, and AC = 10. Find the missing lengths. (14 points) 4. (70 points) Under the letters "a" through "g" in the following table, answer "yes" or "no" to the following questions for each of the given quadrilaterals. a. Are opposite sides congruent and parallel? b. Are opposite angles congruent? c. Are the diagonals always congruent? d. Do the diagonals bisect each other? e. Are the diagonals always perpendicular to each other? f Are all angles congruent? g. Are any two consecutive sides congruent? a b c d e f g Trapezoid Square Rhombus Rectangle Parallelogram 33
34 Quadrilaterals Answers to Test on Quadrilaterals 1. (6 points) In a parallelogram ABCD, if the measurement of angle B exceeds the measurement of angle A by 50, find the measure of angle B. Solution: Let x = the measure of angle A and let x + 50 = the measure of angle B Since two consecutive angles of a parallelogram are supplementary, the measure of angle A plus the measure of angle B equals 180. That is, x + x + 50 = 180 so Since x = 65, x + 50 = 115 and the measure of angle B is (10 points) Given: ABCD is a parallelogram. E is the midpoint of AB. F is the midpoint of DC. Prove: EBFD is a parallelogram. Solution: Statements Reasons 1. ABCD is a parallelogram. 1. Given 2. Segments AB and DC are congruent. 2. Opposite sides of a parallelogram are congruent. 3. E is the midpoint of AB. 3. Given. 4. F is the midpoint of DC. 4. Given. 5. Segments EB and DF are congruent. 5. Halves of congruent segments are congruent. 6. Segments EB and DF are parallel. 6. A parallelogram is a quadrilateral two pairs of opposite sides parallel. 7. EBFD is a parallelogram. 7. If one pair of congruent sides of a quadrilateral are both congruent and parallel, the quadrilateral is a parallelogram. 3. In rectangle ABCD, CB = 6, AB = 8, and AC = 10. Find the missing lengths. (14 points) Solution: AD = 6, CD = 8, EC = 5, AE = 5, DE = 5, EB = 5, and DB = (70 points) Under the letters "a" through "g" in the following table, answer "yes" or "no" to the following questions for each of the given quadrilaterals. a. Are opposite sides congruent and parallel? b. Are opposite angles congruent? c. Are the diagonals always congruent? d. Do the diagonals bisect each other? e. Are the diagonals always perpendicular to each other? f. Are all angles congruent? g. Are any two consecutive sides always congruent? 34
35 a b c d e f g Trapezoid no no no no no no no Square yes yes yes yes yes yes yes Rhombus yes yes no yes yes no yes Rectangle yes yes yes yes no yes no Parallelogram yes yes no yes no no no EXTRA CREDIT: Students can do an investigation on the internet entitled: Investigating Properties of Trapezoids which they can find at This requires a graphing calculator. 35
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