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We will be looking at different aspects of geometry like symmetry, analyzing geometric shapes, and more in the different sections that will follow.
1.
Math 126
Chapter 12
Geometric Shapes
2.
Section 12.1 Recognizing Shapes
The van Hiele Theory
Level 0 (Recognition)
At this lowest level, a child recognizes certain shapes holistically (Greek “holos”
means whole) without paying attention to their components.
In other words, a child may recognize a square without knowing the meaning of a
square; in the same way that a child recognizes an avocado without knowing the
definition of an avocado.
Level 1 (Analysis)
The child starts to focus analytically on the parts of a figure, such as its sides and
angles. Component parts and their attributes are used to describe and characterize
figures. Relevant attributes are understood and are differentiated from irrelevant
In other words, a child starts to understand the definition of a square by knowing that
a square must have exactly 4 congruent sides and 4 right angles.
3.
Section 12.1 Recognizing Shapes
The van Hiele Theory
Level 2 (Relationships)
There are two types of thinking at this level. First, a child understands abstract
relationships among figures. For example, a square is both a rhombus and a rectangle.
Second, a child can use informal deductions to justify observations made at level 1.
For instance, a rhombus is also a parallelogram; or a square is also a rectangle.
Level 3 (Deduction)
Reasoning at this level includes the study of geometry as a formal mathematical
system. A student at this level can use logical reasonings to make deductions.
For example, to deduce that the base angles of an isosceles triangle must be congruent.
A student at this level can also understand the notions of mathematical postulates
(axioms) and theorems, such as the Pythagorean theorem.
4.
Section 12.1 Recognizing Shapes
The van Hiele Theory
Level 4 (Axiomatics)
Geometry at this level is highly abstract and does not necessarily involve concrete or
pictorial models. The postulates or axioms themselves become the object of intense,
rigorous study.
Examples are
a. high dimensional geometry, where we cannot make a concrete model of an n-
dimensional sphere or cube.
b. Non- Euclidean geometry, where lines are not really straight in usual sense, and
non-Euclidean lines are called “geodesics”.
This level of study is only suitable for university or even graduate students.
5.
Level 0 (Recognition)
Children at this level will learn to associate the names “triangle, rectangle,
square” with the appropriate shape, without necessarily understanding the
meaning of the words.
They may just recognize a triangle because it is similar to the one they saw
in the classroom. They may not know that triangle is a shape with 3
This is similar to recognizing their teacher or principal.
Consequently, if the shape is drawn upside down, or drawn in different
colors, or in different sizes, the child may not recognize it as a triangle any
6.
To test their comprehension,
children at this level will be asked to separate the above shapes into
3 categories: Triangles, rectangles, and squares.
7.
Similarly, they will be asked to sort out the circles or ovals
from the following diagram.
8.
Identify the triangles, squares, and rhombi in this quilt.
9.
Section 12.2
Analyzing Geometric Shapes – Level 1
At this level, children will start to understand the meaning
of triangles, different types of quadrilaterals, parallel and
perpendicular lines.
The word “Triangle” will be treated as a compound word
“tri-angle”, where “tri” means 3 in Latin; and “angle”
means corners.
The word “Quadrilateral” will also be explained as the
compound word “quadri-lateral”, where “quadri” means 4
in Latin, and “lateral” means side.
(note: “quadra” means “square” in Latin, hence quadratic equation means degree 2 equation.)
10.
Symmetries
In formal terms, we say that an object is symmetric with respect to a given
mathematical operation, if, when applied to the object, this operation does
not change the object or its appearance.
Reflection Symmetry (also called folding symmetry)
A 2D figure has reflection symmetry if there is a line that the figure can be
“folded over” so that one-half of the figure matches the other half
The “fold line” just described is call the figure’s line (axis) of symmetry.
11.
Lines of symmetry for the following common figures.
12.
Rotation Symmetry
A 2D figure has rotation symmetry if there is a point around which the
figure can be rotated, less than a full turn, so that the image matches the
original figure perfectly.
(click to see animation)
This equilateral triangle has 2 (non-trivial) rotation symmetries, 120° and
240° respectively. Since every figure will match itself after rotating 360°,
we do not consider a 360° rotation as a rotation symmetry.
13.
Rotation symmetries of common figures
We don’t count the trivial 360° rotation symmetry here.
Rectangle Square
(1 symmetry) (3 symmetries) Diamond
(1 symmetry)
(1 symmetry) Trapezoid
Regular Pentagon (no symmetry)
(4 symmetries)
14.
Informal Definition of common shapes
Model Abstraction Description
Top of a window Line segment
Open pair of scissors Angle The union of two line
segments with a
common endpoint.
Vertical flag pole Right angle Angle formed by two
lines, one vertical
and one horizontal.
15.
Model Abstraction Description
Railroad tracks Parallel lines Two distinct line
segments which, if
extended in both
directions, never
meet.
16.
Model Abstraction Description
Bike frame Scalene triangle Triangle with 3 sides
of different lengths
Pennant Isosceles triangle Triangle with at least
two sides of the
same length
Yield sign Equilateral triangle Triangle with three
sides of the same
length
Steel frame of a Right triangle Triangle with one
bridge right angle.
17.
Model Abstraction Description
Railing Parallelogram Quadrilateral with
parallel opposite
sides
Car jack Rhombus Quadrilateral with
four sides equal in
length.
Door Rectangle Quadrilateral with
four right angles
Floor tile Square Quadrilateral with
four sides equal in
length and four right
angles.
18.
Given two line segments l and m on a piece of paper, we can fold the paper
about the mid point of l in such a way that l folds onto itself.
Then m is parallel to l if and only if m also folds onto itself or an
extension of itself.
fol
d lin
e
m
l
19.
A diagonal of a polygon is a line segment that connects two non-
adjacent vertices of that polygon.
There are always two diagonals in a quadrilateral;
but they may not be of the same length; and they
may not be perpendicular to each other.
20.
Let P be the point of intersection of l and m. Fold the paper at point P
such that l folds onto itself.
Then l and m are perpendicular if an only if m is on the fold line
fold line
l
m
21.
Section 12.3
Relationships between Geometric Shapes– Level 2
Students at this level will start to notice the relationships among
different types of triangles.
Triangles
Scalene triangles Isosceles triangles
Equilateral triangles
Right triangles
22.
More Names for quadrilaterals
Model Abstraction Description
Kite Kite Quadrilateral with
two non-overlapping
pairs of adjacent
sides with the same
length
Tail of airplane Trapezoid Quadrilateral with
exactly one pair of
parallel sides
Table/desk Isosceles trapezoid Quadrilateral with
exactly one pair of
parallel sides and the
remaining sides are
of the same length
23.
Summary
Quadrilateral
Trapezoid Parallelogram Kite
Isosceles Rectangle Rhombus
Square
24.
parallelograms
rectangles
kites
rhombi
squares
trapezoids
Isosceles
trapezoids
25.
Section 12.4
An Introduction to a Formal Approach to Geometry
- Level 3
In this section, we will learn more technical terms for lines and angles.
We will also introduce the measure of an angle for the purpose of
formally defining right angles, perpendicular lines, acute and obtuse
triangles.
In the end, we will also see the different types of angles associate with
parallel lines.
26.
Points and Lines in a Plane
In formal geometry, the notions of a point and a line are usually left
undefined (a so-called primitive object). Their properties are then
determined by the axioms which refer to them.
Some terminologies
a line collinear points
concurrent lines non-concurrent
lines
27.
Two different given lines L1 and L2 on a plane are said to be parallel if
they will never intersect each other no matter how far they are
Any line L is also said to be parallel to itself.
29.
The ray CD is defined to be the set of all
points of the line CD that are on the same
side of C as point D.
An angle is the union of two rays with a common endpoint.
i d e
s
vertex interior
side
30.
Any angle divides the plane into 3 non-overlapping parts:
a)The angle itself (two rays with the same end point),
b)The interior of the angle,
c)The exterior of the angle.
The interior of an angle is defined to the region R in the plane
with the property that:
“the segment connecting any two points in this region R is
also completely contained in R.”
This property is called the Convex property. If a region is not convex,
it will be called concave.
31.
Convex and Concave Shapes
A figure is convex if a line segment joining any two points inside the
figure lies completely inside the figure.
32.
More Examples
Category 1 Category 2
What is the mathematical property that separates these two categories of shapes?
Answer: Convex property or Concave property.
33.
Angles are measured by a semi-circular device
called a protractor.
The whole circle is divided into 360 equal parts, each
part is defined to have measure one degree (written 1°).
Hence a semi-circular protractor has 180 degrees.
One degree is divided into 60 minutes and one minute is further
divided into 60 seconds.
27 degrees 35 minutes 41 seconds is written as 27°35’41”
Two angles are said to be congruent if they have the same measure.
34.
Names of angles
A straight angle has 180 degrees. (this type of angle has
no interior nor exterior)
An obtuse angle has measure between 90° and 180°.
A right angle has exactly 90°.
An acute angle has measure less than 90°.
35.
Classification of triangles according to their angles.
A triangle with one right angle is called a right triangle.
A triangle with one obtuse angle is called an obtuse triangle.
A triangle with 3 acute angles is called an acute triangle.
36.
Classification of triangles according to their sides.
A triangle with 3
A triangle with 3 different sides is called a
equal sides is called scalene triangle.
an equilateral
triangle.
A triangle with 2 equal
sides is called an
isosceles triangle.
37.
Two angles are called vertical angles if they are opposite to each other
and are formed by a pair of intersecting lines.
A B
Any pair of vertical angles are always congruent.
38.
More special angles
Two angles are said to be supplementary if their measures add up to 180°.
α β
Two angles are said to be complementary if their measures add up to 90°.
α
β
39.
Perpendicular Lines
Two lines are said to be perpendicular to each other if they intersect to
form a right angle
40.
Angles associate with Parallel Lines
Given two line L1 and L2 (not necessarily parallel) on the plane, a third line T
is called a transversal of L1 and L2 if it intersects these two lines.
L1
L2
T
41.
Let L1 and L2 be two lines (not necessarily parallel) on the plane, and T
be a transversal.
a) a and form a pair of corresponding angles.
b) c and form a pair of corresponding angles etc.
L1
a
c
L2
T
42.
Let L1 and L2 be two lines (not necessarily parallel) on the plane, and T
be a transversal.
c) c and form a pair of alternate interior angles.
d) d and form a pair of alternate interior angles.
L1
c
d L2
T
43.
Let L1 and L2 be two lines (not necessarily parallel) on the plane, and T
be a transversal.
e) a and form a pair of alternate exterior angles.
f) b and form a pair of alternate exterior angles.
L1
a
L2
T
44.
Let l and m be two lines (not necessarily parallel) on the plane,
and T be a transversal.
1 and 3 are called interior angles on the same side of the
The above lines l and m are parallel if and only if the measure of any
pair of interior-angles-on-the-same-side-of-the-transversal add up to
45.
Let L1 and L2 be two lines on the plane, and T be a transversal.
If L1 and L2 are parallel, then
a) any pair of corresponding angles are congruent,
b) any pair of alternate interior angles are congruent,
c) any pair of alternate exterior angles are congruent.
L1
L2
T
46.
Let L1 and L2 be two lines on the plane, and T be a transversal.
a) if there is a pair of congruent corresponding angles, then L1 and L2
are parallel.
b) if there is a pair of congruent alternate interior angles, then L1 and L2
are parallel.
c) if there is a pair of congruent alternate exterior angles, then L1 and L2
are parallel.
L1
L2
T
47.
Angle Sum in a Triangle
Draw an arbitrary triangle on a
piece of paper and label all 3
b angles.
Next cut out the triangle, and
then cut it into 3 parts (as
indicated by the dashed lines)
Arrange the 3 angles side by
a side, can you get a straight
c angle?
Conclusion:
The angle sum in a triangle is always 180°
48.
Angle Sum in a Quadrilateral
If we draw a diagonal, then
the quadrilateral will be cut
into two triangles. Hence
the angle sum of a
quadrilateral must be twice
the sum of angles in a
triangle, i,e, 360.
49.
Application of Degree Measure
Angles can be used to indicate directions. The only difference
is that the measure can be greater than 180º.
In navigation, the direction can be any value between 0º and
50.
The Bearing System
The exact (magnetic) North is defined to be 0 degree.
Any other direction is defined to be the number of
degrees away from exact North measuring in the
clockwise direction.
N
130º
south east direction
51.
The Bearing System
In particular, 90º is equal to exact East,
N
90º = East
52.
The Bearing System
and 180º is equal to exact South,
N
180º = South
53.
The Bearing System
and 270º is equal to exact West,
N
270º = West
54.
Runway Numbers
In any airport, each runway is assigned a number
according to the direction it is pointing at – except
that the units digit is omitted for simplicity.
For example, runway 24 is actually pointing at 240º,
and it means that during final approach, the aircraft is
heading 240º - which is about south west.
55.
This is one of the many signs that you will see in a big commercial airport. It
tells the pilots which runway is in front of them.
56.
San Diego International
58.
Section 12.5
Regular Polygons, Tessellations,
and Circles
59.
Basic Definitions
A simple closed curve in the plane is a curve that can be traced with the
same starting and stopping points and without crossing or retracing any
part of the curve.
A simple closed “curve” that is made up with line segments is called a
A polygon where all sides are congruent and all angles are congruent is
called a regular polygon.
60.
The word "polygon" derives from the Greek poly, meaning "many",
and gonia, meaning "angle".
Equilateral Square Regular
triangle n=4 pentagon
n=3 n=5
Regular Regular Regular
hexagon heptagon octagon
n=6 n=7 n=8
61.
Polygons and their nomenclature
A Triangle (from Latin) has 3 sides
A Quadrilateral (from Latin) has 4 sides (tetra is from Greek)
A Pentagon (from Greek) has 5 sides
A Hexagon (from Latin) has 6 sides
A Heptagon (from Greek) (or a Septagon from Latin?) has 7 sides
62.
In fact, “Septagon” is not an official word for the 7-gon, it is not even in a
dictionary. It was invented by some elementary school teachers to make it
easier to remember. The Latin word septem means 7 and September means
the seventh month.
The old Roman calendar began the year in January, (named after the Roman
god of fortune, Janus), and September was the seventh month. Afterwards,
Julius Augustus (46 BC) named two more then-29 day periods after himself
and September came to be as we know it in the Gregorian Calendar, the ninth
63.
An Octagon (from Greek) has 8 sides
A Nonagon (from Latin) has 9 sides.
A Decagon (from Greek) has 10 sides.
A polygon with more than n (>10) sides is usually just
called an n-gon.
65.
Angle Sum in other Polygons
What is the sum of all angles in a quadrilateral?
Answer: 180 × 2 = 360
What is the sum of all angles in a pentagon?
Answer: 180 × 3 = 540
66.
Angle Sum in other Polygons
For a polygon with n sides, the angle sum is
(n – 2) × 180°
67.
Angles in a polygon
In a regular pentagon:
the measure of a central angle is 360°/5 = 72°
the measure of an exterior angle is also 360°/5 = 72°
the measure of a vertex angle is 180° – 72° = 108°
68.
Angles & Angle Sums in Regular
polygons
For a regular pentagon,
360
m(central angle) = 5 72
le
ng
la
ra
nt
ce
m(vertex angle)
center = (3 × 180) ÷ 5
= 108
vertex angle
69.
Tessellations (or Tilings)
A tessellation is an arrangement of congruent shapes that cover an
entire area with no overlaps or gaps.
A 2D geometric figure R is said to tessellate (or tile) the plane if the entire
plane can be completely covered by (an infinite number of) congruent
copies of R with no overlaps or gaps.
70.
A regular tessellation means a tessellation made up of congruent
regular polygons.
A tessellation with congruent copies of several different regular polygons
are called semiregular tessellations
71.
It is also possible to tile a plane with congruent copies of several different
irregular polygons, such as below.
72.
Convex and Concave Polygons
a convex quadrilateral a concave (i.e. non-
convex) quadrilateral
A polygon X is said to be convex if you take any two points on X (including
the boundary), the line segment joining them lies entirely within the tile
(again including the boundary).
73.
Question: What kind of polygons can tessellate the
1. Can triangles tessellate the plane?
Any triangle can tessellate the plane.
2. Can quadrilaterals tessellate the plane?
Any square can tessellate the plane.
Any rectangle can tessellate the plane.
Any convex quadrilateral can tessellate the plane.
In fact, any quadrilateral (including non-convex ones) can
tessellate the plane.
3. Can regular pentagons tessellate the plane?
No, a regular pentagon will not tessellate the plane.
4. Can hexagons tessellate the plane?
Any regular hexagon can tessellate the plane.
In fact, exactly 3 classes of convex hexagons can tile the plane.
(this was proved by K. Reinhardt in his 1918 doctoral thesis.
See next slide.)
74.
Type 2 Type 3
Type 1
A + B + D = 360° A = C = E = 120°
B + C + D = 360°
C + E + F = 360° a = a'
A + E + F = 360°
a=d c = c'
c=e e = e'
Regular hexagon
belongs to Type 3
75.
8. He then went on to explore the tessellations by irregular but convex
pentagons and found 5 classes that do tile the plane.
He felt that he had found all of them even though he could not give
a proof because he claimed that it would be very tedious to do so.)
76.
In 1968, after 35 years working on the problem on and off, R. B. Kershner,
a physicist at Johns Hopkins University, discovered 3 more classes of
pentagons that will tessellate.
Kershner was sure that he had found all of them, but again did not offer a
complete proof, which “would require a rather big book.”
Shortly after an article of the “complete” classification of convex pentagons
into 8 types appeared in Scientific American (July 1975), an amateur
mathematician (R. James III) discovered a 9th type!
Between 1976 and 1977, a San Diego housewife Marjorie Rice, without
formal education in mathematics beyond high school, found 4 more types!
A 14th type was found by a mathematics graduate student in 1985.
In 2015, University of Washington Bothell mathematicians Casey Mann,
Jennifer McLoud, and David Von Derau discovered a 15th regular tiling
with convex pentagon using a computer algorithm.
Yet, no one can say that this is the complete list.
(the 15 types of pentagons that tile the plane)
77.
The elevator lobby on the 7th floor
of the Mathematics building in the
Ohio State University.
Note that the tiles are irregular
pentagons, and yet they tessellate
the plane.
78.
Columnar Basalts -
Naturally occurring hexagonal columns
that tessellate the plane.
79.
With the situation so intricate for convex pentagons, you might think that it
must be even worse for polygons with 7 or more sides. However, the
situation is remarkably simple, as Reinhardt proved in 1927:
A convex polygon with 7 or more sides cannot tessellate.
80.
Circles
A Circle is the set of all points in the plane that are at a fixed distance
from a given point called the center.
The distance from any point on the circle to the center is called the
radius of the circle. Any segment connect the center to the edge is also
called a radius.
The length of any line segment whose endpoints are on the circle and
which contains the center is called the diameter of the circle. The
segment is also called a diameter of the circle.
81.
Circles
Circles have the following 3 properties that make them very useful.
1. They are highly symmetrical, hence they have a sense of beauty and
are often used in designs due to aesthetical reasons.
eg. dinnerware.
However, plates
don’t have to be
83.
Circles
Circles have the following 3 properties that make them very useful.
1. They are high symmetrical, hence they have a sense of beauty and
are often used in designs.
eg. dinnerware.
However, plates
don’t have to be
84.
2. A circle has a center, and every point on a circle bears the same distance
from the center. This is called the constant radius property.
Applications: wheels and gears.
Note: An ellipse also has a center, but the radius varies in length
according to its placement.
85.
3. For a given (fixed) perimeter, the circle has the largest area.
Applications: water bottles, soda cans, and any container for
pressurized liquid are all cylindrical in shape.
87.
Section 12.6
Describing 3-Dimensional Shapes
(This will be taught after section 13.2)
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