Download DAY-4---Quadrialaterals-RM-10

Day 4
Explorations of Quadrilaterals
Title
Parallelism and Perpendicularity
Goal
Explore the concept of parallelism and perpendicularity through the
exploration of parallelograms and alternate interior angles.
Standard
Addressed
3MG 2.3; 4MG3.1; 4MG3.3; 4MG3.8; 5MG2.1; 5MG2.2; 6MG2.1;
6MG2.2
3.G.CA-1; 4.G..1; 5GCA-1; 5G.CA.2; 7.G.5
Materials for
Teacher
Over head, handouts, color pencils, scissors, and rulers
Materials for
Students
Lesson handouts, color pencils, scissors, and rulers
Description
Discover the notions of parallelism and perpendicularity for the special
quadrilaterals: parallelograms, rectangles, and squares. Identify attributes
of these quadrilaterals.
Reflection
Looking Ahead
Parallelism and perpendicularity are basic notions in geometry.
Understanding their special properties is a crucial point for further study
of trapezoids, rhombuses and kites, and 3D figures later on.
Link to text
We discover the notions of parallelism and perpendicularity for the special quadrilaterals:
parallelograms, rectangles, and squares. Parallelism is one of the most important notions in the
historical context of Euclidean geometry, and it deserves special attention. We prepare the way in
studying more general quadrilaterals and polygons in the next lectures.
Opening Activity:
What is a quadrilateral in general? What is a convex quadrilateral? What is a concave
quadrilateral?
1
Definition. The figure that is enclosed by four arbitrary segments is called a quadrilateral. It is
formed by “gluing” two triangles on a common side. The picture above defines a convex
quadrilateral, which means it contains all the segments with ends on its sides.
The figure above has one convex quadrilateral (left) and one concave quadrilateral (right). A
concave quadrilateral does not contain all the segments with ends on its sides.
Area. The area of a quadrilateral is the sum of the areas of the two triangles formed by two
adjacent sides and a diagonal. We will come back to this issue later on.
2
A Classification of Quadrilaterals
Using the definitions we have seen so far, construct a hierarchy showing the relationships among
the following figures: Polygons, Quadrilaterals, Parallelograms, Rhombuses, Rectangles,
Squares, Kites, and Isosceles Trapezoids.
3
Activity:
What is a parallelogram? What are special properties of a parallelogram? How can you “ensure”
that the opposite sides are parallel?
Definition. The figure that is enclosed by two pairs of parallel lines is called a parallelogram. It
is a figure that has four sides so that the two opposite sides are parallel and the other two sides
are parallel to each other.
Questions:
1. What angles in the parallelogram below have the same measure and why?
2. Why are the opposite sides equal to each other?
Answers: To respond to these two questions and to similar questions we will encounter about
other quadrilaterals it helps to understand relationships created by parallel lines and a transversal.
Definition: A transversal is a line that intersects a set of n lines in exactly n different points. In
the diagram below, GH is a transversal of the set of lines (AB, CD, EF).
When a transversal intersects two lines, eight angles are formed as shown below. Certain pairs of
these angles have special names.
4
•
Angles I and L, and also angles J and K, are called alternate interior angles.
•
Angles G and N, and also angles H and M, are called alternate exterior angles.
•
Angles G and K, angles I and M, angles H and L, and angles J and N are called
corresponding angles.
•
Angles G and J, angles H and I, angles K and N, and angles L and M are called vertical
angles.
5
Angle Relationships for Parallel Lines
1.
2.
3.
4.
5.
Cut out the extra triangle provided for you on the last page of this handout.
Use color pencils to shade the three angles of the triangle using different colors.
Tessellate the area between the parallel lines below using your cut triangle.
Mark all the angles on the tessellation appropriately using the color pencils.
Discuss the angle relationships shown by your tessellation.
6
Examples: Using parallel lines to find angle measures
a. Use the figure below to find the measures of the angles specified. Justify your answer for
each angle.
mB = _______ Why?
mBDF = _______ Why?
mADF = _______ Why?
mAFD = _______ Why?
mDFE = _______ Why?
mC = _______ Why?
b. Find the measures of the numbered angles.
m1 = _______
m2 = _______
m3 = _______
m4 = _______
m1 = _______
7
Back to the parallelogram: What kind of lines do you see in this picture? How many angles are
there? Which angles have the same measure?
2. Answer to: Why the opposite sides of a parallelogram are equal to each other? What other
important properties of a parallelogram can we discover?
D
A
B
C
 The triangles ABC and CDA are congruent.
 The triangles OBC and ODA are congruent. Same for the pair of triangles OBA and
ODC. This ensures that the diagonals of a parallelogram cross in their midpoint.
 Fact: Consecutive angles are supplementary.
 Fact: Diagonals bisect each other.
 Fact: Opposite angles are congruent.
8
Rhombus Exploration
Rhombus. A parallelogram with all sides equal to each other. It follows that the diagonals are
perpendicular to each other (why?)
A bit of history: Euclidean geometry is a mathematical system attributed to the Alexandrian
Greek mathematician Euclid, whose Elements is the earliest known systematic discussion of
geometry. For over two thousand years, the adjective “Euclidean” was unnecessary because no
other sort of geometry had been conceived. Euclid's axioms (rules) seemed so intuitively obvious
that any theorem proved from them was deemed true in an absolute, often metaphysical, sense.
Today, however, much non-Euclidean geometry is known, the first ones having been discovered
in the early 19th century. One important application of the notion of non-Euclidean geometry is
Einstein's theory of relativity.
The most controversial rule Euclid spelled out in his Elements is the Parallel Postulate
historically known as the Fifth Postulate: “That, if a straight line falling on two straight lines
make the interior angles on the same side less than two right angles, the two straight lines, if
produced indefinitely, meet on that side on which are the angles less than the two right angles.”
To the ancients, the parallel postulate seemed less obvious than the others. Euclid himself seems
to have considered it as being qualitatively different from the others, as evidenced by the
organization of the Elements: the first 28 propositions he presents are those that can be proved
without it. Many alternative axioms can be formulated that have the same logical consequences
as the parallel postulate. For example Playfair's axiom states: “Through a point not on a given
straight line, at most one line can be drawn that never meets the given line. “
Problems to Explore
9
1. Calculate the other angles inside the parallelogram.
2. Draw a parallelogram that has a 60º angle. Calculate its other angles.
3. Draw a rhombus that has a 45º angle and a side of 2 inches. Calculate its other angles.
Calculate its perimeter.
4. If a rhombus has one angle of 55º, can you determine its other angles?
5. Can you cut one triangle out of a rhombus and then glue it back (not in the same place :)
in order to make a square?
10
11