HS/Geometry
Mathematics
Unit: 11
Lesson: 01
Duration: 7 days
Quadrilaterals
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
G.2B
G.5
G.5B
G.7
G.7B
G.7C
Geometric structure The student analyzes geometric relationships in order to make
and verify conjectures.
Make conjectures about angles, lines, polygons, circles, and three-dimensional
figures and determine the validity of the conjectures, choosing from a variety of
approaches such as coordinate, transformational, or axiomatic.
Geometric patterns The student uses a variety of representations to describe
geometric relationships and solve problems
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.
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.
Use slopes and equations of lines to investigate geometric relationships, including
parallel lines, perpendicular lines, and special segments of triangles and other
polygons.
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
© 2007, TESCCC
•
•
•
•
•
rhombus
rectangle
square
consecutive angles
opposite sides
Revised 10/01/07
•
•
consecutive sides
opposite angles
page 1 of 50
HS
Geometry
Unit: 11 Lesson: 01
Materials:
•
•
colored paper
patty paper
•
•
•
•
protractor
chart paper
markers
straightedge
Resources:
Advance Preparation:
1.
2.
3.
4.
5.
6.
Handout: Quadrilaterals and Angle Sums (one copy per student)
Handout: Quadrilaterals and Angle Sums Practice (one copy per student)
Handout: All in the Family (one copy per student)
Handout: Properties of Quadrilaterals (one copy per student)
Handout: Quadrilaterals and Coordinate Geometry (one copy per student)
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
Notes for Teacher
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.
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 pre-assess the level
of understanding of the students.
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.
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
HS
Geometry
Unit: 11 Lesson: 01
Instructional Procedures
EXPLAIN 1
Notes for Teacher
1. Debrief the previous EXPLORE activity and facilitate a discussion of
student findings using the questions below.
Facilitation Questions:
• 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.
• 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
1. Distribute copies of All in the Family to students.
2. Have students complete the All in the Family with a partner or in groups.
SUGGESTED DAY 2
MATERIALS
• handout: Quadrilaterals and Angle
Sums Practice (one copy per
student)
SUGGESTED DAY 2 AND 3
MATERIALS
• 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
1. Debrief the previous EXPLORE activity with students. Be sure to facilitate
a discussion regarding common properties of certain quadrilaterals.
Facilitation Questions:
• Which quadrilaterals had no pair of opposite sides parallel? The
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.
• Which quadrilaterals had characteristics of a rectangle? The
square.
• Which quadrilaterals had characteristics of a rhombus? The
square.
• Can you give examples of how can this information could be used
to classify quadrilaterals? Explain. Rectangles, rhombi, and squares
are all unique quadrilaterals yet they are related. Each has their own
unique characteristics but they all have the characteristics of a
parallelogram. A square is a type of rectangle because it satisfies the
defining characteristics of a rectangle. etc.
© 2007, TESCCC
Revised 10/01/07
SUGGESTED DAY 3 AND 4
MATERIALS
• handout: Properties of
Quadrilaterals (one copy per
student)
TEACHER NOTE
The Practice Problems from
Properties of Quadrilaterals provide
students with an opportunity to
demonstrate their knowledge of
quadrilaterals and their properties, and
albeit brief, demonstrate their ability to
prove aspects of quadrilaterals based
on their properties. Should the teacher
wish to include more deductive proofs, a
suggestion would be to resort to proving
page 3 of 50
HS
Geometry
Unit: 11 Lesson: 01
Instructional Procedures
Notes for Teacher
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.
any of the numerous properties
discovered in the previous EXPLORE
phase as time permits.
ELABORATE
SUGGESTED DAY 4, 5 AND 6
MATERIALS
• handout: Quadrilaterals and
Coordinate Geometry (one copy
per student)
1. Distribute Quadrilaterals and Coordinate Geometry to students.
2. Have students complete Quadrilaterals and Coordinate Geometry.
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
1. Distribute Trace the Ancestry to students.
2. Have students complete Trace the Ancestry.
SUGGESTED DAY 7
MATERIALS
• 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
HS
Geometry
Unit: 11 Lesson: 01
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 four-sided 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…
z
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
Geometry.
© 2007, TESCCC
Revised 10/01/07
page 5 of 50
HS
Geometry
Unit: 11 Lesson: 01
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=95o
m ∠ ABC=104o
m ∠ 8= 18º
m ∠ 9= 115º
m ∠ BED=80o
AC ⊥ CD
m ∠ 10= 60º
m ∠ 11= 82º
m ∠ 1=83o
m ∠ 2=62o
m ∠ 12= 100º
m ∠ 13= 97º
m ∠ 3=101o
m ∠ 4=104o
m ∠ 14= 93º
m ∠ 15= 75º
m ∠ 5=46o
m ∠ 6=70o
m ∠ 16= 35º
m ∠ 17= 17º
m ∠ 18= 32º
m ∠ 19= 14º
m ∠ 7=35o
B
16
8
A
19
9
13
5
C
3
1
2
E
10
12
4
11
14
7
G
© 2007, TESCCC
15
6
D
Revised 10/01/07
17
18
F
page 6 of 50
HS
Geometry
Unit: 11 Lesson: 01
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 four-sided 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.
z
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
HS
Geometry
Unit: 11 Lesson: 01
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=95o
m ∠ ABC=104o
m ∠ 8=
m ∠ 9=
m ∠ BED=80o
AC ⊥ CD
m ∠ 10=
m ∠ 11=
m ∠ 1=83o
m ∠ 2=62o
m ∠ 12=
m ∠ 13=
m ∠ 3=101o
m ∠ 4=104o
m ∠ 14=
m ∠ 15=
m ∠ 5=46o
m ∠ 6=70o
m ∠ 16=
m ∠ 17=
m ∠ 18=
m ∠ 19=
m ∠ 7=35o
B
16
8
A
19
9
13
5
C
3
1
2
E
10
12
4
11
14
7
G
© 2007, TESCCC
15
6
D
Revised 10/01/07
17
18
F
page 8 of 50
HS
Geometry
Unit: 11 Lesson: 01
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 ≅ ∠C and ∠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
HS
Geometry
Unit: 11 Lesson: 01
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 .
sur
sur
suur
suur
b. Use your pencil to draw 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
HS
Geometry
Unit: 11 Lesson: 01
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
R
a. Find the measure of the angles of the rhombus. Record your answers below.
See student measures…
suur
sur
sur
b. Let SR be a transversal for SP and RQ . What conclusions can you make about lines
suur
sur
SP and RQ based on your knowledge of lines and transversals? Explain.
sur suur
∠PSR and ∠SRQ are supplementary; therefore, SP RQ .
suur
sur
sur
c. Let SP be a transversal for SR and PQ . What conclusions can you make about lines
suur
sur
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
HS
Geometry
Unit: 11 Lesson: 01
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
HS
Geometry
Unit: 11 Lesson: 01
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
B
1
2
4
3
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.
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?
suur
suur
M and N are the midpoints of sides AD and 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
HS
Geometry
Unit: 11 Lesson: 01
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 ≅ ∠R and ∠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
HS
Geometry
Unit: 11 Lesson: 01
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
S
a. A special type of trapezium is a kite. A kite is a trapezium with two pairs of congruent
adjacent sides. The figure below is a kite.
K
E
I
T
b. Use a piece of patty paper and a straightedge to trace the kite. Label the vertices of the
kite. Use a straightedge to draw the diagonals of the kite on your patty paper tracing.
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
with I. Examine the crease
sur
in your paper. What does this verify about diagonal EI ?
KT bisects EI .
© 2007, TESCCC
Revised 10/01/07
page 15 of 50
HS
Geometry
Unit: 11 Lesson: 01
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
HS
Geometry
Unit: 11 Lesson: 01
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.
suur
sur
sur
suur
b. Use your pencil to draw 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
HS
Geometry
Unit: 11 Lesson: 01
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
R
a. Find the measure of the angles of the rhombus. Record your answers below.
suur
sur
sur
b. Let SR be a transversal for SP and RQ . What conclusions can you make about lines
suur
sur
SP and RQ based on your knowledge of lines and transversals? Explain.
suur
sur
sur
c. Let SP be a transversal for SR and PQ . What conclusions can you make about lines
suur
sur
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
HS
Geometry
Unit: 11 Lesson: 01
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
HS
Geometry
Unit: 11 Lesson: 01
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
HS
Geometry
Unit: 11 Lesson: 01
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
HS
Geometry
Unit: 11 Lesson: 01
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
S
a. A special type of trapezium is a kite. A kite is a trapezium with two pairs of congruent
adjacent sides. The figure below is a kite.
K
E
I
T
b. Use a piece of patty paper and a straightedge to trace the kite. Label the vertices of the
kite. Use a straightedge to draw the diagonals of the kite on your patty paper tracing.
c. Use a protractor to measure the angles formed by the intersection of the diagonals.
Record the measures on your patty paper tracing. What conclusion can you make
about the diagonals of the kite?
d. Fold and crease your patty paper tracing so that E coincides
with I. Examine the crease
sur
in your paper. What does this verify about diagonal EI ?
© 2007, TESCCC
Revised 10/01/07
page 22 of 50
HS
Geometry
Unit: 11 Lesson: 01
Properties of Quadrilaterals (KEY)
Properties of convex quadrilaterals:
ƒ Have four sides.
ƒ Have four vertices and angles.
ƒ Sum of the angles equals 360o.
ƒ 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.
ƒ Trapezium – a quadrilateral with no pairs of parallel sides
o Kite – two congruent pairs of adjacent sides
ƒ Property• Diagonals are perpendicular.
• One of the diagonals bisects the other.
ƒ Trapezoid – a quadrilateral that has only one pair of parallel sides
o Isosceles trapezoid – non parallel legs are congruent
ƒ Property• The base angles of an isosceles trapezoid are congruent.
ƒ Parallelogram – a quadrilateral with two pair of parallel sides, opposite sides are parallel
ƒ 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
ƒ Property• The diagonals of a rectangle are congruent.
o Rhombus – parallelogram with four congruent sides
ƒ 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
HS
Geometry
Unit: 11 Lesson: 01
Properties of Quadrilaterals (pg 2) KEY
2. Write a proof for the statement, “The diagonals of a rhombus are perpendicular.”
R
H
B
M
© 2007, TESCCC
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).
O
Revised 10/01/07
page 24 of 50
HS
Geometry
Unit: 11 Lesson: 01
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 = 29.
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
P
y
T
4
E
3
C
x = 4, y = 3, TE = 5, RC = 5, RP = EP = TP = CP = 2.5
© 2007, TESCCC
Revised 10/01/07
page 25 of 50
HS
Geometry
Unit: 11 Lesson: 01
Properties of Quadrilaterals (pg 4) KEY
5. Given the formula for the area of a triangle, Atriangle = 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
d2
d1
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
1
triangles. The legs of each triangle are d1 and d 2 ; and, the area of one of the triangles is
2
2
1 1
1
1
( d1 )( d 2 ) = d1d 2 . Therefore, the area of all four triangles (the rhombus) is
2 2
2
8
1
1
4( d1d 2 ) = d1d 2 .
8
2
© 2007, TESCCC
Revised 10/01/07
page 26 of 50
HS
Geometry
Unit: 11 Lesson: 01
Properties of Quadrilaterals (pg 5) KEY
6. Suppose a carpenter is framing a rectangular room and wants to verify that the 4-sided 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
HS
Geometry
Unit: 11 Lesson: 01
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 = 1200 , m∠P = 600 , and m∠R = 1000
T
R
(12x+60)º
(5y)º
(4x+40)º
80º
P
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 (20 + 5 2) ≈ 27.1 ft.
K
45º
45º
E
I
60º
30º
T
10. Use the information in the trapezoid below to find HK, IJ, m∠KHI , m∠KJI and m∠HIJ .
HK = 12, IJ= 12, m∠KHI = 1200 , m∠KJI = 600 , m∠HIJ = 1200
H
I
6 3
60º
K
© 2007, TESCCC
J
Revised 10/01/07
page 28 of 50
HS
Geometry
Unit: 11 Lesson: 01
Properties of Quadrilaterals
Properties of convex quadrilaterals:
ƒ Have four sides.
ƒ Have four vertices and angles.
ƒ Sum of the angles equals 360o.
ƒ 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.
ƒ Trapezium – a quadrilateral with no pairs of parallel sides
o Kite – two congruent pairs of adjacent sides
ƒ Property• Diagonals are perpendicular.
• One of the diagonals bisects the other.
ƒ Trapezoid – a quadrilateral that has only one pair of parallel sides
o Isosceles trapezoid – non parallel legs are congruent
ƒ Property• The base angles of an isosceles trapezoid are congruent.
ƒ Parallelogram – a quadrilateral with two pair of parallel sides, opposite sides are parallel
ƒ 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
ƒ Property• The diagonals of a rectangle are congruent.
o Rhombus – parallelogram with four congruent sides
ƒ 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
HS
Geometry
Unit: 11 Lesson: 01
Properties of Quadrilaterals (pg 2)
2. Write a proof for the statement, “The diagonals of a rhombus are perpendicular.”
R
H
B
M
© 2007, TESCCC
O
Revised 10/01/07
page 30 of 50
HS
Geometry
Unit: 11 Lesson: 01
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
P
y
T
© 2007, TESCCC
4
Revised 10/01/07
E
3
C
page 31 of 50
HS
Geometry
Unit: 11 Lesson: 01
Properties of Quadrilaterals (pg 4)
5. Given the formula for the area of a triangle, Atriangle = 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
d2
© 2007, TESCCC
d1
Revised 10/01/07
page 32 of 50
HS
Geometry
Unit: 11 Lesson: 01
Properties of Quadrilaterals (pg 5)
6. Suppose a carpenter is framing a rectangular room and wants to verify that the 4-sided 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
HS
Geometry
Unit: 11 Lesson: 01
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
R
(12x+60)º
(5y)º
(4x+40)º
80º
P
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
I
60º
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
60º
K
© 2007, TESCCC
J
Revised 10/01/07
page 34 of 50
HS
Geometry
Unit: 11 Lesson: 01
Quadrilaterals and Coordinate Geometry (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 = 2.5 2
BE = 2.5 2
EC = 2.5 2
ED = 2.5 2
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
HS
Geometry
Unit: 11 Lesson: 01
Quadrilaterals and Coordinate Geometry (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
HS
Geometry
Unit: 11 Lesson: 01
Quadrilaterals and Coordinate Geometry (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 = 2.5 5
BE = 2.5 5
EC = 2.5 5
ED = 2.5 5
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
HS
Geometry
Unit: 11 Lesson: 01
Quadrilaterals and Coordinate Geometry (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
HS
Geometry
Unit: 11 Lesson: 01
Quadrilaterals and Coordinate Geometry (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 ≈ 143.1°
m∠CDA ≈ 143.1°
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
HS
Geometry
Unit: 11 Lesson: 01
Quadrilaterals and Coordinate Geometry (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
HS
Geometry
Unit: 11 Lesson: 01
Quadrilaterals and Coordinate Geometry
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
HS
Geometry
Unit: 11 Lesson: 01
Quadrilaterals and Coordinate Geometry (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
HS
Geometry
Unit: 11 Lesson: 01
Quadrilaterals and Coordinate Geometry (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
HS
Geometry
Unit: 11 Lesson: 01
Quadrilaterals and Coordinate Geometry (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
HS
Geometry
Unit: 11 Lesson: 01
Quadrilaterals and Coordinate Geometry (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
HS
Geometry
Unit: 11 Lesson: 01
Quadrilaterals and Coordinate Geometry (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
HS
Geometry
Unit: 11 Lesson: 01
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
Kites
Trapezoids
Isosceles
Trapezoid
Parallelograms
Rectangle
Rhombus
Square
© 2007, TESCCC
Revised 10/01/07
page 47 of 50
HS
Geometry
Unit: 11 Lesson: 01
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.
b.
c.
d.
Graph the figure, labeling points. See student samples…
Name the quadrilateral. Square
Find the length of each side.
4 2
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 .
Rectangle-The Square has 4 right angles.
Rhombus-The Square has 4 congruent sides.
Parallelogram-The Square has opposite sides parallel since consecutive angles
are supplementary.
© 2007, TESCCC
Revised 10/01/07
page 48 of 50
HS
Geometry
Unit: 11 Lesson: 01
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
HS
Geometry
Unit: 11 Lesson: 01
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