TOPIC 16
quadrilaterals
We will begin this new topic with definitions, as we usually do. Your exercise to begin is to find
the definitions of the quadrilaterals. You can use your textbook or Topic 1, where we had most
of the definitions we needed.
Quadrilateral:
Parallelogram:
Rhombus:
Rectangle:
Square:
Trapezoid:
Isosceles trapezoid.
Kite:
Topic 14: Quadrilaterals
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Inscriptable quadrilateral:
Let’s make some drawing of quadrilaterals, using these definitions and what we have also
noted about triangles.
1.
Draw a parallelogram (This is not a construction.) Mark what is true based on the
definition.
2.
Draw a rhombus.
3.
Draw a trapezoid.
4.
Draw an isosceles trapezoid.
5.
Draw a kite.
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6.
Draw an equilateral quadrilateral which is not equiangular.
7.
Draw an equiangular quadrilateral which is not equilateral.
8.
Draw an equilateral and equiangular quadrilateral.
9.
Draw a quadrilateral with exactly three congruent sides.
10.
Draw a circle with your compass. Draw an inscriptable quadrilateral which is scalene (all
non-congruent sides, if possible). Measure its angles.
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Is there a hierarchy of the kinds of quadrilaterals?
11.
Is a rectangle a rhombus? (always, sometimes, never)
12.
Is a square a rhombus?
13.
Is a parallelogram a rectangle?
14.
Is a trapezoid a rhombus?
15.
Is there an order in which each of the classifications of quadrilaterals are sub-categories
of other categories? See if you can conjecture one? (This means, for example, all
parallelograms are quadrilaterals, and all rectangles are parallelograms, and all squares
are ……..)
We will do a Sketchpad investigation of the properties of each kind of quadrilaterals. This will
be a homework exercise, and you will come back to class ready to make some educated
conjectures about the sides, angles, and diagonals of the various quadrilaterals.
We will start with parallelograms. Remember that the definition is not a quality that has to be
investigated.
Homework:
We will use the Geometer’s Sketchpad to do the investigation.
16.
Open up a new sketch.
CONSTRUCT a line.
Mark two points, and with TEXT, name them A and B on the line.
SELECT a new point D, not on the line.
Topic 14: Quadrilaterals
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CONSTRUCT a line parallel to AB through D.
CONSTRUCT a line parallel to AD through B.
Locate the point of intersection of these last two lines, and call that point C.
Now, we want to make line segments and HIDE the lines:
SELECT A and B, and CONSTRUCT a segment.
SELECT A and D, and CONSTRUCT a segment.
SELECT B and C, and CONSTRUCT a segment.
SELECT C and D, and CONSTRUCT a segment.
Then one by one, select each full line, and HIDE it. This should leave just the
parallelogram.
Draw a copy of your sketch here.
17.
Measure each of the sides of the parallelogram. Record and label them here.
18.
Grab one of the points and move it to a new place. Note the four side measures. Record
and label them again.
19.
Make a conjecture about the lengths of the sides of a parallelogram.
20.
Measure the four angles of the parallelogram. Record and label them here.
21.
Grab one of the points and move it to a new place. Note the four angles measures.
Record and label them again.
22.
Make a conjecture about the measures of the angles of a parallelogram.
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23.
CONSTRUCT the two diagonals. Mark the point of intersection, and label it E.
24.
Measure the diagonals. Are they congruent? Could they be congruent by contorting the
parallelogram. Describe the results.
25.
Do the diagonals bisect each other? (Check more than one version…i.e., contort your
figure a few times… of your figure to make your decision.)
26.
The diagonals cut each of the four angles of the parallelogram into pairs. Measure the
adjacent pairs, and determine whether the diagonals bisect any of the angles of the
parallelogram.
27.
Contort the parallelogram, and look at whether the diagonals bisect any of the angles
Describe your results.
28.
When the diagonals intersect, they create angles. Describe the angles that occur where
they intersect (do the angle meet at right angles, or not?).
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Now, you will work in teams of 2-3 to do deductive proofs to see whether your conjectures about
the properties of the parallelograms are always true.
We will list the conjectures here, and the proofs will go in your notes:
Conjecture 1: If a quadrilateral is a parallelogram, then ____________________________.
Conjecture 2: If a quadrilateral is a parallelogram, then ____________________________.
Conjecture 3: If a quadrilateral is a parallelogram, then ____________________________.
Conjecture 4: If a quadrilateral is a parallelogram, then ____________________________.
Conjecture 5: If a quadrilateral is a parallelogram, then ____________________________.
Use this space to do some of the proofs of he conjectures.
Topic 14: Quadrilaterals
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Next, look at the new theorems we have accomplished that begin with IF A QUADRILATERAL
IS A PARALLELOGRAM, then ……….
Which of these has a true converse? Or in other words, what are necessary and sufficient criteria
for a quadrilateral to be a parallelogram. We will make some guesses, and then work to see
whether each of the guesses is true or not. There could be a criterion which is not a converse of
one of the theorems we just proved, and one or more of the theorems might not have a true
converse:
Conjecture 1: If_________________________, then a quadrilateral is a parallelogram.
Conjecture 2: If_________________________, then a quadrilateral is a parallelogram.
Conjecture 3: If_________________________, then a quadrilateral is a parallelogram.
Conjecture 4: If_________________________, then a quadrilateral is a parallelogram.
Conjecture 5: If_________________________, then a quadrilateral is a parallelogram.
Conjecture 6: If_________________________, then a quadrilateral is a parallelogram.
We will have teams of 2-3 try to prove (or disprove) each one of these, and come to a conclusion
about the truth of each of them.
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Let’s do investigations about the properties of the other quadrilaterals. Again, this is a
homework assignment that you will do with the Geometer’s Sketchpad.
Now, we will investigate a rectangle. Write the definition of a rectangle here.
29.
You are on your own to use the definition to construct a rectangle with Sketchpad.
Remember that you need to HIDE the lines and keep the line segments.
30.
Measure each of the sides of the rectangle. Record and label them here.
31.
Grab one of the points and move it to a new place. Note the four side measures. Record
and label them again.
32.
Make a conjecture about the lengths of the sides of a rectangle.
33.
Measure the four angles of the rectangle. Record and label them here. (You actually
know the result here.)
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34.
Grab one of the points and move it to a new place. Note the four angles measures.
Record and label them again.
35.
Make a conjecture about the measures of the angles of a rectangle.
36.
CONSTRUCT the two diagonals. Mark the point of intersection, and label is E.
37.
Measure the diagonals. Are they congruent? Contort the rectangle. Describe the results.
38.
Do the diagonals bisect each other? (Check more than one version…i.e., contort your
figure a few times… of your figure to make your decision.)
39.
The diagonals cut each of the four angles of the rectangle into pairs. Measure the
adjacent pairs, and determine whether the diagonals bisect any of the angles of the
parallelogram.
40.
Contort the rectangle, and look at whether the diagonals bisect any of the angles.
Describe your results.
Topic 14: Quadrilaterals
41.
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When the diagonals intersect, they create angles. Describe the angles that occur where
they intersect.
We will continue the process above with other special kinds of quadrilaterals.
Use Sketchpad to investigate a rhombus, square, trapezoid, isosceles trapezoid, and kite. You
are looking at the lengths of sides, parallelism of sides, lengths of diagonals, angle between
diagonals, bisection, and whether interior angles are bisected by the diagonals.
42.
After you have completed your investigation, complete the chart below with the
properties of each kind of quadrilateral. (Note that some of the “properties” are
contained in the definitions, and other “properties” can be verified, so they have theorem
status.)
Congruent
sides
Parallel
sides
Length of
diagonals
Angle
between
diagonals
Diagonals Interior
bisected
angles
bisected by
diagonals
Parallelogram
rectangle
rhombus
square
trapezoid
kite
Remember your work with coordinate geometry? We learned how to find the slope of a
segment, the length of a segment using the coordinates, determining whether segments were
perpendicular or parallel using slopes, calculated midpoints, and solved two linear equations to
find where lines intersected.
Let’s suppose we want to look at some quadrilaterals using coordinates. We will see whether we
can determine the same characteristics as above, but this time we will use coordinate geometry
methods. (Your proofs above are called “synthetic proofs”, and here they are called “coordinate
proofs”. The results should be the same, but the method is very different.
Topic 14: Quadrilaterals
43.
44.
45.
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We know that a rectangle has 4 right angles and opposite sides which are parallel.
Record your answers as the questions are asked. Provide evidence (that is, show how
you used the distance formula or slopes or midpoints.)
a.
Suppose that two of the coordinates of a rectangle are A(0 , 0) and B( 6 , 0).
b.
Write two other possible coordinates.
c.
Calculate the lengths of the diagonals. Compare them. Are they congruent?
d.
Are the diagonals perpendicular?
a.
Suppose that two of the coordinates of a rectangle are A(0 , 0) and B( 6 , 4).
b.
Write two other possible coordinates.
c.
Calculate the lengths of the diagonals. Compare them. Are they congruent?
d.
Are the diagonals perpendicular?
We know that a rhombus has 4 congruent sides and opposite sides which are parallel.
Record your answers as the questions are asked. Provide evidence (that is, show how
you used the distance formula or slopes or midpoints.)
a.
Suppose that three of the coordinates are A(0 , 0), B( 5, 0) and D(3 , 4).
b.
Locate the coordinates of C.
Topic 14: Quadrilaterals
46.
47.
48.
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c.
Calculate the lengths of the diagonals. Compare them. Are they congruent?
d.
Are the diagonals perpendicular?
One diagonal of rhombus PQRS has endpoints P(1 ,2) and R(6,4).
a)
Name the coordinates of the other two vertices. (not a unique answer)
b)
Find the lengths of the diagonals.
One diagonal of parallelogram MNTW has endpoints N(5 , -1) and W(0 . 6).
a)
Name the coordinates of the other two vertices. (not a unique answer)
b)
Find the lengths of the diagonals.
Give the coordinates of an isosceles trapezoid whose longest base sits on the positive end
of the x-axis.
a)
What are the lengths of the two diagonals?
b)
Are the diagonals perpendicular? Give evidence?
Topic 14: Quadrilaterals
c)
49.
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If your answer to (b) is YES, are the diagonals always perpendicular? If your
answer to (b) is NO, could the diagonals be perpendicular in a different isosceles
trapezoid?
Give the coordinates of a trapezoid which has two right, interior angles.
a)
Do this once is the bases are horizontal.
b)
Do this again if none of the sides are horizontal or vertical.
We will choose various of the attributes which we believe to be true about the quadrilaterals, and
we will prove that they are true. We will sometimes use a synthetic (deductive) proofs, and
sometimes coordinate proofs.
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Here are some numerical problems to solve, using the attributes that we have established.
Angles in Quadrilaterals
Write the measures of all of the angles in each quadrilateral.
1.
3.
2.
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4.
5.
6.
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Quadrilateral Numerical Problems
(Hand-in)
Provide the measures for all numbered angles and (lower case) lettered sides of these
quadrilaterals. Think about the properties of each quadrilateral to guide your conclusions.
1.
2.
3.
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4.
5.
6.
7. AGF
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Topic 14: Quadrilaterals
8.
9.
10.
Rectangle ABCD
m<2 = 30
m<8 = 45
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Quadrilaterals inside quadrilaterals.
When the consecutive midpoints of various kinds of quadrilaterals are connected, a new
quadrilateral is formed. See if you can determine the kind of quadrilateral that is formed in each
case. Give the most specific kind of quadrilateral possible. You should be able to prove that
your answer is correct.
1.
2.
3.
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4.
5.
6.
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7.
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