1. GEOMETRY

2. GEOMETRY GEOMETRY is about measuring GEO means “Earth”
METRY means “Measure”
GEOMETRY is also about shapes, their properties and relationships

3. Optical Illusions through art using geometric concepts
GEOMETRIC ILLUSIONS
1.1
Optical Illusions through art using geometric concepts

4. GEOMETRIC ILLUSIONS 1.1 Penrose triangle
The Penrose triangle, also known as the Penrose tribar, is an impossible object.
The impossible cube or irrational cube is an impossible object that draws upon the ambiguity present in a Necker cube illusion. An impossible cube is usually rendered as a Necker cube in which the edges are apparently solid beams. This apparent solidity gives the impossible cube greater visual ambiguity than the Necker cube, which is less likely to be perceived as an impossible object. The illusion plays on the human eye’s interpretation of 2-dimensional as 3-dimensionall objects.

5. GEOMETRIC ILLUSIONS
1.1
A famous perceptual illusion in which the brain switches between seeing a young girl and an old woman

6. GEOMETRIC ILLUSIONS
1.1
The Kanizsa triangle is an optical illusions first described by the Italian psychologist Gaetano Kanizsa in In the accompanying figure a white equilateral triangle is perceived, but in fact none is drawn. This effect is known as a subjective or illusory contour. Also, the nonexistent white triangle appears to be brighter than the surrounding area, but in fact it has the same brightness as the background.

7. GEOMETRIC ILLUSIONS
The Necker cube is an ambiguous line drawing. It is a wire-frame drawing of a cube in isometric perspective, which means that parallel edges of the cube are drawn as parallel lines in the picture. When two lines cross, the picture does not show which is in front and which is behind. This makes the picture ambiguous; it can be interpreted two different ways. When a person stares at the picture, it will often seem to flip back and forth between the two valid interpretations (so-called multistable perception).

8. 1.3 Identifying Congruent and Similar Figures
Congruent means that the figures must have same shape and size
Similar means figures must have same shape, but can also have same size

9. 1.4
Exploring Symmetry
A figure has Line Symmetry or Reflection Symmetry if it can be divided into 2 parts, each of which is the mirror image of the other. Some figures have 2 or more lines of symmetry. a. b. c.
This lobster figure has a horizontal line of symmetry.
This saucer figure has a vertical and horizontal line of symmetry
This ying/yang figure has 2 congruent parts, but it has no line of symmetry
Although this ying/yang figure lacks line symmetry, it does have Rotational Symmetry.
To be rotational a figure must coincide with itself after rotating 180⁰ or less, either clockwise or counter-clockwise.

10. 1.4 Exploring Symmetry Reflective Symmetry Rotational Symmetry
Line of Symmetry
Mirror image of the other side of line of symmetry
Must coincide with itself after rotating 180 ⁰or less

11. ( ) ( ) ( ) 1.4 The Midpoint Formula M = x1 + x2 , y1 + y2 • •
The midpoint of the line segment joining A (x1, y1) and B (x2, y2) is as follows:
C ( 1, 3) •
M = (
x1 + x2 ,
y1 + y2 ) 2
B ( 4, 1)
M = (
─2 + 4 ,
5 + 1 ) 2
Each coordinate of M is the mean of the corresponding coordinates of A and B.
M = ( 2 , 6 )

12. To understand GEOMETRY you must first learn to speak the language by studying the terminology
Next you need to understand the logic of geometry through deductive and inductive reasoning
Then you need to practice applying learned principles in new problem situations
Finally you need to remember learned concepts as building blocks for new ones to follow

13. The Building Blocks of Geometry
UNDEFINED TERMS : can’t be defined by simpler terms. [ Point , Line, Plane ]
DEFINED TERMS : can be defined by the undefined terms or previously defined terms so it is easier to describe geometric figures and relationships
POSTULATES : a statement that is accepted without proof.
THEOREMS : a statement that is proven to be true.
The progression of undefined, defined terms, postulates and theorems lead to evolving new logical generalizations.

14. Coordinate & Noncoordinate Geometry
1.5
Coordinate & Noncoordinate Geometry
Coordinate Plane
y axis
Quadrant II
( – , + )
Quadrant I
( + , + )
( 0 , 0 )
x axis
Quadrant III
( – , – )
Quadrant IV
( + , – )
x coordinate is first
y coordinate is second
Ordered pairs in form of ( x , y )

15. 1.5 Slope and Rate of Change • •
SLOPE of a non-vertical line is the ratio of a vertical change (RISE) to a horizontal change (RUN).
Slope of a line:
m = y2 – y1 = RISE
x2 – x1 = RUN
(differences in y values)
(differences in x values) y
( x2 , y2 ) • 
y2 – y1
RISE
( x1 , y1 ) •
x2 – x1
RUN x

16. 1.5 Slope and Rate of Change
CLASSIFICATION OF LINES BY SLOPE
A line with a + slope rises from left to right [ m > 0 ]
Positive Slope
A line with a – slope falls from left to right [ m < 0 ]
Negative Slope
A line with a slope of 0 is horizontal [ m = 0 ]
0 Slope
A line with an undefined slope is vertical [ m = undefined, no slope ]
No Slope

17. 1.5 Slope and Rate of Change m1 = m2 m1 = - 1 m2 or m1 m2 = - 1
SLOPES OF PARALLEL & PERPENDICULAR LINES
PARALLEL LINES: the lines are parallel if and only if they have the SAME SLOPE.
m1 = m2
PERPENDICULAR LINES: the lines are perpendicular if and only if their SLOPES are NEGATIVE RECIPROCALS.
m1 = - 1 m2 or
m1 m2 = - 1

18. Slope and Rate of Change
Slope of a line:
m = y2 – y1 = RISE (differences in y values)
x2 – x1 = RUN (differences in x values)
1.5
Slope and Rate of Change
Ex 1: Find the slope of a line passing through ( – 3, 5 ) and ( 2, 1 )
Let ( x1, y1 ) = ( – 3, 5 ) and ( x2, y2 ) = ( 2, 1 )
m = – 1 = 4
– 3 – – 5 OR
m = – 5 = – 4 5 •
( - 3, 5 )
– 4 • x
( 2, 1 ) y

19. Coordinate & Noncoordinate Geometry
(also known as Analytic Geometry)
Shows a graph
Non-Coordinate Geometry
(also known as Euclidean Geometry)
has no graph

20. 1.6 Perimeter & Area ● Square Rectangle
P = 4 s and A = s2 P = 2 l + 2 w and A = l w
Triangle Circle
P = a + b + c and A = ½ b h C = 2 ∏ r and A = ∏ r2 w l S S r a c ● h b

21. Find the Perimeter & Area
1.6
Find the Perimeter & Area
Example 1
Each square on the grid at the left is 1 foot by 1 foot. Find the perimeter and area of the green region.
SOLUTION: The distance around the shaded figure is 40 units. Thus the perimeter of the region is 40 feet. By counting the shaded squares, you can find that the area is 36 square feet.
Can you think of another way to find the area?
Subtract the number of non-green units within the 6 x 8 = 48 total units.
Or, 48 – 12 = 36

22. Area of larger rectangle
1.6
Finding an Area
6 ft
What is the area of the sidewalk?
6 ft
40 ft
Area of sidewalk
Area of larger rectangle
Area of pool = ─
= ( 36 ) ( 52 ) – ( 24 ) ( 40 )
= – 960
= 912 square feet
24 ft

23. 2.1 Undefined Terms Undefined Terms Description Notation Point
A point indicates position; it has no length, width, or depth
• A
A point is named by a single capital letter
Line
A line is a set of continuous points that extend indefinitely in either direction
A B
A line is identified by naming two points on the line and drawing a line over the letters. ↔ AB
Plane
A plane is a set of points that forms a flat surface that has no depth and that extends indefinitely in all directions
A plane is usually represented as a closed four-sided figure and is named by placing a capital letter at one of the corners. • • P

24. 2.1
This illustration shows that lines may lie in different planes or in the same plane. B L P ● B ● A k
Here lines L and AB are on plane “B” while lines k and AB are on plane “P”. NOTE: line AB lies in both “B and P” planes, which is also a intersecting line of these planes.

25. 2.1 Defined Terms Defined Terms Description Illustration Line segment
A line segment is a part of a line consisting of two points, called end points, and the set of all points between them.
Notation: A B
Ray
A Ray is a part of a line consisting of a given point, called the end point and the set of all points on the one side of the end point.
A Ray is always named by using two points, the first of which must be the end point. The arrow above always points to the right.
Notation: LM
Line
Or Opposite Rays
Opposite rays have the same end point and form a line.
The line is indicated by KB or BK
KX and KB are opposite rays. • • A B ● ● L M ● ● ● X K B

26. 2.1 Defined Terms Defined Terms Description Illustration Angle
An angle is the union of two rays having the same end point. The end point is called the vertex of the angle, and the rays are called the sides of the angle.
The measure of < A is denoted by m < A < 90. Angles are classified as acute, right, obtuse and straight.
Acute
0 ⁰ < m < A < 90 ⁰
2 angles are adjacent if they share a common vertex and side, but have no common interior points
Right
m < A = 90 ⁰
Obtuse
90 ⁰ < m < A < 180 ⁰
Straight
m < A < 180 ⁰
Reflex
180 ⁰ < m < A < 360 ⁰ J ● ● ● K L
Vertex: K
Sides KJ and KL 3 2
< 1 and < 2 are adjacent
< 1 and < 3 are not adjacent ● 1 V

27. 2.1 Naming Angles ) An angle may be named in one of three ways:
Using three letters, the center letter corresponding to the vertex of the angle and the other letters representing points on the sides of the angle. For example, in Figure 1, the name of the angle whose vertex is T can be angle RTB ( < RBT )
2. Placing a number of the vertex and in the interior of the angle. The angle may then be referred to by the number. For example, in Figure 2, the name of the angle whose vertex is T can be < 1 or < RTB or < BTR.
Figure Figure 2 . R R ● ●
Exterior of angle
Interior of angle ) 1 ● ● T ● T ●
Exterior of angle B B

28. ) ) ) 2.1 Naming Angles An angle may be named in one of three ways:
Using a single letter that corresponds to the vertex, provided that this does not cause any confusion. There is no question which angle on the diagram corresponds to angle A in figure 3, but which angle on the diagram is angle D?
Actually 3 angles are formed at vertex D:
1. angle ADB ( < ADB )
2. angle CDB ( < CDB )
3. angle ADC ( < ADC) . B C
In order to uniquely identify the angle having D as its vertex, we must either name the angle using three letters or introduce a number into the diagram. ) ) ) A D
Figure 3

29. Example of Naming Angles
2.1
Example of Naming Angles
Use three letters to name each of the numbered angles in the accompanying diagram. M B C 3 E 2
< 1 =
< 2 =
< 3 =
< 4 = 1 4 A D L

30. Example of Naming Lines and Line Segments
2.1
Example of Naming Lines and Line Segments ● ● ● J W R
Name the line in three different ways =
Name three different segments =
Name four different rays =
Name a pair of opposite rays =

31. 2.1
Definitions
The purpose of a definition is to make the meaning of a term clear. A good definition must:
Clearly identify the word or expression that is being defined
State the distinguishing characteristics of the term being defined, using commonly understood or previously used
Be expressed in a grammatically correct sentence
Example: consider the terms collinear and non-collinear points
Collinear points are points that lie on the same line
Non-collinear points are points that do not lie on the same line ● S ● ● ● C ● R B ● A T
Much of geometry involves building on previously discussed ideas.
Example: we can use current knowledge of geometric terms to arrive at a definition of a triangle. How would you draw a triangle? If you start with 3 non-collinear points and connect them with line segments, a triangle is formed.

32. 2.1
Definitions
A good definition must be reversible as shown in the following table
Definition
Reverse of the Definition
Collinear points are points that lie in the same line.
Points that lie on the same line are collinear points.
A right angle is an angle whose measure is 90 ◦
An angle whose measure is 90 ◦ is a right angle.
A line segment is a set of points.
A set of points is a line segment.
The first two definitions are reversible since the reverse of the definition in a true statement. The reverse of the third definition is false since the points may be scattered as in example to right. ● ● ● ● ● ●