Properties of a parallelogram, a rectangle, a rhombus, a square

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 1-56822-067-7
1

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

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

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

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

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

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

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

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

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.
What do you notice
17. Drag the vertices.
What happens to the lengths and angle measures What do you notice
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.
What do you notice
24. Drag the vertices.
What happens to the lengths and angle measures What do you notice
10

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.
What do you notice
Opposite sides and opposite angles are congruent.
Consecutive angles are supplementary.
17. Drag the vertices.
What happens to the lengths and angle measures What do you notice
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.
What do you notice
The diagonals bisect each other. Alternate interior angles are congruent
24. Drag the vertices.
What happens to the lengths and angle measures What do you notice
Although the lengths and angle measures change, the diagonals still bisect
each other, and the alternate interior angles are still congruent.
11

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

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

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

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 shown on
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

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.
What do you notice
14. Drag the vertices.
What happens to the lengths and angle measures What do you notice
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.
What do you notice
21. Drag the vertices.
What happens to the lengths and the angle measures What do you notice

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.
What do you notice
13. Drag the vertices.
What do you notice
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.
What do you notice
21. Drag the vertices.
What do you notice
17

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.
What do you notice
8. Drag the vertices.
What do you notice
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.
What do you notice
16. Drag the vertices.
What do you notice
18

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.
What do you notice
Opposite sides are equal in length.
Each angle is ninety degrees.
14. Drag the vertices.
What happens to the lengths and angle measures What do you notice
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.
What do you notice
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 What do you notice
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

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.
What do you notice
All four sides are congruent.
Opposite angles are congruent
Any two consecutive angles are supplementary.
13. Drag the vertices.
What do you notice
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.
What do you notice
The diagonals bisect each other.
The diagonals bisect its angles.
The diagonals are perpendicular to each other.
21. Drag the vertices.
What do you notice
The diagonals still bisect each other.
The diagonals still bisect its angles.
The diagonals are still perpendicular to each other.
20

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.
What do you notice
All four sides are congruent.
All four angles are congruent.
Each angle is 90 degrees.
8. Drag the vertices.
What do you notice
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.
What do you notice
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.
What do you notice
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

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.

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

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.
What do you notice
12. Drag the vertices.
What do you notice
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.
What do you notice
20. Drag the vertices.
What do you notice
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.
What do you notice
27. Drag the vertices.
What do you notice

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.
What do you notice
12. Drag the vertices.
What do you notice
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.
What do you notice
20. Drag the vertices.
What do you notice
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.
What do you notice
27. Drag the vertices.
What do you notice
25

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.
What do you notice
Adjacent angles are supplementary.
12. Drag the vertices.
What do you notice
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.
What do you notice
The diagonals are not congruent and they do not bisect each other.
20. Drag the vertices.
What do you notice
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.
What do you notice
The median is parallel to the bases and equal to one-half of their sum.
27. Drag the vertices.
What do you notice
The median is still parallel to the bases.
26

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.
What do you notice
The base angles are congruent. The legs are congruent.
12. Drag the vertices.
What do you notice
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.
What do you notice
The diagonals are congruent. The shorter segments of each diagonal are
congruent and the longer segments are congruent
20. Drag the vertices.
What do you notice
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.
What do you notice
The median is parallel to the bases.
27. Drag the vertices.
What do you notice
The median is parallel to the bases.
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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.
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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.
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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
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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

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.
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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
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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 115.
2. (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 = 10
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 always congruent
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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
http://www.ti.com/calc/docs/act/92geo1.htm. This requires a graphing calculator.
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