Download Lesson 13: Parallelograms - Logo-Math

Parallelograms
Lesson 13: Parallelograms
Purpose: This lesson introduces the definitions and properties of parallelograms. The Logo
program provides an excellent environment to explore different types of parallelograms.
This lesson uses a procedure with three variables to efficiently draw a given parallelogram.
This lesson contains optional information on how to set the dimensions of the graphic window.
Essential Question: Can you apply your knowledge of interior and exterior angle measures to
code and investigate the properties of parallelograms?
Definitions:
A quadrilateral is a closed polygon with exactly four sides.
A parallelogram is a quadrilateral that has two pairs of parallel sides.
A rectangle is a parallelogram with a right angle.
A square is a rectangle with all sides of equal length (equilateral).
A rhombus is a parallelogram with all sides of equal length (equilateral).
Recall that a regular polygon is both equiangular and equilateral. The square (introduced in
Lesson #5) is the only regular quadrilateral.
Note that the precise wording of a definition will vary depending on the book you are using. For
example, a square could be defined as a rhombus with a right angle.
LESSON 13 – TASK 1:
The graphic in Figure 13.1 shows parallelogram ABCD with given side lengths of 135 pixels by
195 pixels and an acute interior angle of 40 degrees. An auxiliary line was drawn from point A
perpendicular to line BC. Write and test the Logo code for this shape starting at point A. (check
your work at the end of this lesson).
Figure 13.1
A parallelogram provides a good example of pairs of lines that are cut by a transversal. For
example, lines AB and DC are cut by transversal AD (and transversal BC). Lines AD and BC
are cut by transversal AB (and transversal DC). It helps to use colored pencils when locating a
transversal line!
KG SHAFER, ©2012 revision date – March 8, 2013
Parallelograms
Definitions and Theorems
Vertical Angles are defined as two non-adjacent angles formed by the intersection of two
distinct lines. This definition was introduced in Lesson #4.
Vertical Angles Theorem
Vertical angles have the same measure (are congruent).
Alternate Interior Angles are located between the pair of lines cut by the transversal but on
opposite sides of the transversal.
Alternate Interior Angle Theorem (AIA)
If a pair of alternate interior angles are congruent, then the lines are parallel.
If a pair of parallel lines are cut by a transversal, then the alternate interior angles are
congruent.
*The two conditional statements of the AIA Theorem can be written as one bi-conditional
statement. See the end of this lesson for more information.
LESSON 13 - TASK 2:
In Figure 13.2, the measure of angle BAD is 40°. Mark each of the remaining 40-degree angles
with a single arc tic mark. An example of the single arc tic mark is shown at angle BAD. Arc tic
marks of the same type indicate angle measures that are equal. After marking the diagram,
complete the chart below.
In the first row of the chart, write the vertical angle pairs for all of the 40° degree angles.
In the second row, write the alternate interior angle pairs for all of the 40° angles.
Figure 13.2
Vertical Angle
Pairs
∠BAD and
∠MAN
Alternate
Interior Angle
Pairs
∠BAD and
∠TBA
KG SHAFER, ©2012 revision date – March 8, 2013
Parallelograms
Types of Parallelograms
Figure 13.3 illustrates six parallelograms drawn on a grid (each section is 100 x 100). The first
two parallelograms are rectangles that are not squares. The side dimensions are 50 x 80 pixels.
The third shape is an 80 x 80 square. The next two shapes are parallelograms that are not
rectangles or rhombi. Their side dimensions are 50 x 80. The last shape is an 80 x 80 rhombus
that is not a square. The acute angle in the last three shapes is 60 degrees. It is helpful to
create shapes that have certain restrictions when you are studying parallelograms.
Figure 13.3
In this lesson, I will use one procedure, titled ‘PARA’, to efficiently draw each of the six shapes
shown above. When you write a program with variables, you must make an assignment for
each variable. In the PARA procedure, I made the assignments as follows:
X = length of the first side drawn
Y = length of the second side drawn (adjacent side)
A = measure of the base angle at the lower left corner of the shape
LESSON 13 – TASK 3:
Open the editor and name the new procedure PARA (as shown in figure 13.4).
Type X Y A for the input variables
Enter REPEAT 2[FD X RT (180-A) FD Y RT A) in the command window
Define (save) the procedure
Figure 13.4
Run the procedure from the listener window by typing para 80 50 90 without any brackets. This
is the code for the first rectangle in figure 13.3. Note that when you run the procedure from the
listener window you must type three values after the PARA command.
KG SHAFER, ©2012 revision date – March 8, 2013
Parallelograms
A. Use the Logo code in the third column of the chart below to explore results of the PARA
procedure. Try to predict what shape will be drawn by examining the information given in the
first two columns.
B. Complete the chart by indicating if your Logo shape matches the definition for each
quadrilateral. An example is shown for the first row.
Label each row with the most descriptive category i-iv.
i.
Parallelogram that is not a rhombus or rectangle
ii.
Rhombus that is not a square
iii.
Rectangle that is not a square
iv.
Square
Angle Constraint Sample Code Parallelogram Rectangle Square Rhombus Label
a=90
x<y
Para 50 80 90
Yes
Yes
No
No
iii
a=90
x>y
Para 80 50 90
a=90
x=y
Para 80 80 90
a< 90
x<y
Para 50 80 60
a< 90
x>y
Para 80 50 60
a< 90
x=y
Para 80 80 60
C. Complete each sentence with the missing word Always, Sometimes or Never. You can
provide a sketch to the right of the sentence.
i.
Parallelograms are _______________ quadrilaterals.
ii.
Squares are _______________ parallelograms.
iii.
Rhombi are _______________ squares.
iv.
Squares are _______________ rectangles.
v.
Rectangles are _______________ squares.
vi.
Squares are _______________ rhombi.
vii.
Rhombi are _______________ rectangles.
viii.
Rectangles are _______________ quadrilaterals.
KG SHAFER, ©2012 revision date – March 8, 2013
Parallelograms
D. Complete the following charts by indicating which properties of diagonals are true or false for
each given shape. Use Logo to help you justify your decision (no guessing!). If you do not
know what a line of symmetry is, you may look it up on the Internet or in the glossary of a
geometry book.
Parallelogram that is not a Rectangle
or a Rhombus
Diagonals are
congruent
false
Rectangle that is not a Square
Diagonals are
congruent
Diagonals bisect
each other
Diagonals bisect
each other
Diagonals are lines
of symmetry
Diagonals are lines
of symmetry
Rhombus that is not a square
Square
Diagonals are
congruent
Diagonals are
congruent
Diagonals bisect
each other
Diagonals bisect
each other
Diagonals are lines
of symmetry
Diagonals are
lines of symmetry
true
E. A Venn diagram can be used to show the relationship between sets of items. Label the
regions of the Venn diagram shown in Figure 13.5 with the types of parallelograms listed below.
Let the universal set be Polygons.
Squares
Rectangles
Rhombi
Quadrilaterals
Parallelograms
Figure 13.5
KG SHAFER, ©2012 revision date – March 8, 2013
Parallelograms
Lesson Thirteen Project
Sketch three different quadrilaterals that are a:
a) Parallelogram that is not a Rectangle or a Rhombus
b) Rectangle that is not a Square
c) Rhombus that is not a square
Measure one angle of each parallelogram and two adjacent
sides. Then, write/test the logo code for that parallelogram.
Do not use measurements that were used in this lesson.
Solution for Lesson 13 - Task 1
rt 50 fd 135 rt 40 fd 195 rt 140 fd 135 rt 40 fd 195
Discussion of a bi-conditional statement
An if-then statement is called a conditional statement. If you have two inverse if-then
statements (in the form of if hypothesis then conclusion and if conclusion then hypothesis)
that are true, they can be combined into one bi-conditional statement. The format for the biconditional statement is to use “if and only if” between the two conditions. The AIA Theorem
can be written in either of the following statements.
A pair of alternate interior angles are congruent if and only if the lines are parallel.
Or
A pair of parallel lines are cut by a transversal if and only if the alternate interior angles are
congruent.
KG SHAFER, ©2012 revision date – March 8, 2013
Parallelograms
Optional
I used the setwinsize “graphics command (introduced below) in the program “Myhouse (with
grass)” from Lesson 8. Different screen sizes and/or resolution settings can distort what is seen
when a Logo program is executed. The command winsize “graphics will return the current
setting. The default screen settings for three of my computers are listed here:
MacBook
1280 x 800 resolution
winsize “graphics
Result: [1125 437]
MacBook Pro
1440 x 900 resolution
Winsize “graphics
Result: [618 462 ]
Windows 7 PC with a large monitor
1920 x 1080
winsize “graphics
Result: [1742 604]
New Commands:
The Logo Coordinate Grid has a default size (dimensions) that depends on the computer you
are using. Also, the size of the graphics window can be changed by dragging the edges. To
standardize the graphics window’s dimensions, use the setwinsize “graphics command.
setwinsize “graphics [width height] needs two inputs, the width and the height in pixels.
In order to create a larger turtle, use the setturtlesize command. Note that a larger turtle size is
helpful when working with younger students or with students who have a visual impairment.
I used four commands to create the graphic shown in Figure 13.6
gridon
cs
setturtlesize 3
setwinsize “graphics [800 400]
Figure 13.6
KG SHAFER, ©2012 revision date – March 8, 2013