Good Afternoon!

1
Good Afternoon!
Today we will be learning about Review of
Geometry
Lets warm up
The dimensions of Rectangular prisms are
given. Find their volume.
1) Height 5 cm Width 5 cm Depth 5 cm
2) Height 4 cm Width 3 cm Depth 6 cm
1) 125 cm3
2) 72 cm3
3) Height 8 cm Width 10 cm Depth 9 cm
4) Height 7 cm Width 7 cm Depth 7 cm
3) 720 cm3
4) 343 cm3
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Points, lines, segments, rays
Geometry is all about shapes and their
properties.
The two most common subjects in geometry are
1) Plane Geometry
2) Solid Geometry
Plane geometry is the study of plane figures in
the plane such as points, lines, line segments,
rays, angles, circles, triangles, quadrilaterals,
and other polygons ... shapes that can be drawn
on a piece of paper.
Solid Geometry is the study of three dimensional
objects like cubes and pyramids. It is called
three-dimensional, or 3D because there are three
dimensions width, depth and height.
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Point
X
A point is a location in space.

• A point is an exact location .

• Points are dimensionless,
• i.e., a point has no width, length, or height.

We locate points relative to some arbitrary
standard point, often called the "origin".
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Line
DE
D
E
A line is a group of points on a straight path
that extends to infinity.
Any two points on the line can be used to name
it. This line is called line DE.

• Its length, having no limit, is infinite.

• It has no width or height.

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Line segment
XY
X
Y
A line segment is a part of a line that has two
end points. A line segment is the path of
shortest distance between two points.
The two end points of the line segment are used
to name the line segment. This line segment is
called segment XY.
All the points "between" the two points make up a
line segment.
A line segment has one dimension, length. It has
no width or height.
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Ray
OP
O
P
A ray is part of a line. A ray extends
indefinitely in one direction, but ends at a
single point in the other direction. That point
is called the end-point of the ray.
A ray is named starting with its end point first
and then any other point on the ray second.
This ray is called ray OP.
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Using the graphic figure
1) VY
1) Name a line.
2) Name a line segment with U as an end point.
2) UT
3) VY
3) Name a ray with V as an end point.
4) Name a line segment with X as an end point.
4) XU
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Perpendicular, parallel intersecting lines
Lines are parallel if they are always the same
distance apart (called "equidistant"), and will
never meet.
l
m
Lines m and l are parallel lines.
They will travel to infinity in either direction
and never intersect.
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Intersecting lines
Two or more lines that meet at a point are called
intersecting lines. That point would be on each
of these lines.
x
Q
y
In the Figure, lines x and y are intersecting
lines and intersect at point Q.
Lines can only intersect at one point and only
one point.
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Perpendicular lines.
If the line segments meet or cross each other to
form square corners, they are perpendicular to
each other.
s
right angles
t
The little box drawn in the corner, means "at
right angles.
Perpendicular lines intersect at a point and form
4 right angles.
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Symbols in Geometry
Here are the some geometrical symbols
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Now you try!
Classify each pair of lines as parallel,
intersecting, or perpendicular.
1)
2)
2) intersecting
1) parallel
4)
3)
4) parallel
3) perpendicular
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Angles (right, acute, obtuse) protractor
What Is an Angle?
An angle is a combination of two rays with a
common endpoint.
B
angle AOB
O
vertex
A
arm
The endpoint (O) is known as the vertex of the
angle And the rays (OA and OB) are called the
sides or arms of the angle .
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Angles On a Straight Line
If we know one angle is 45, what is angle x" ?
x
45
Angle x will be 180 - 45  135
This method can be used to find angles on one
side of a straight line.
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Angles Around a Point
Angles around a point will always add up to 360
degrees.
110
40
60
150
The angles here all add to 360.
40 110 150 60 360
Because of this, if there is an unknown angle we
can always find it.
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Complementary Angles
Two Angles are Complementary if they add up to 90
degrees (a Right Angle).
60
30
These two angles (40 and 50) are Complementary
Angles, because they add up to 90.
But the angles don't have to be together to
Complement each other.
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Supplementary Angles
Two Angles are Supplementary if they add up to
180 degrees (a Straight Angle).
120
60
These two angles (120 and 60) are Supplementary
Angles, because they add up to 180.
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Now you try!
Find the Complement of the following
1) 57
2
1
?
59
2) 31
33
?
Find the Supplement of the following
4) 45
3) 60
3
4
120
135
?
?
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Triangles (isosceles, equilateral, right)
A triangle is one of the basic shapes of
geometry A polygon with three corners or
vertices and three sides or edges which are line
segments.
angle ACB Or angle c
vertex
arm
The three angles always add to 180.
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Interior Angle An Interior Angle is an angle
inside a shape.
Exterior Angle The Exterior Angle is the angle
between any side of a shape, and a line extended
from the next side.
Exterior Angle
135
Interior Angle
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If you add up the Interior Angle and Exterior
Angle you get a straight line, 180.
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Triangle Classification
The basic elements of any triangle are its sides
and vertices. Triangles are classified depending
on relative sizes of their elements.
Triangles can be classified according to their
internal angles.
Acute Triangle An acute triangle is a triangle
whose angles are all acute (i.e. less than 90).
In the acute triangle shown above, a, b and c are
all acute angles.
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Right Triangle A right triangle is a triangle
with a right angle (i.e. 90).
The side opposite the right angle is always the
triangle's longest side. It is called the
hypotenuse of the triangle.
The other two sides are called the legs.
hypotenuse
leg
right angle
leg
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Obtuse Triangle An obtuse triangle has one
obtuse angle (i.e. greater than 90º).
The longest side is always opposite the obtuse
angle.
In the obtuse triangle shown above, a is the
obtuse angle.
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Types of Triangles
There are three special names given to triangles
that tell how many sides (or angles) are equal.
The triangle classification is summarized as
follows
Equilateral Triangle An equilateral triangle has
all three sides equal in length. Its three angles
are also equal and they are each 60º.
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Isosceles Triangle An isosceles triangle has two
sides of equal length. The angles opposite the
equal sides are also equal.
Scalene Triangle A scalene triangle has no sides
of equal length. Its angles are also all
different in size.
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Now you try!
Classify each triangle as Equilateral, Isosceles
or Scalene
1)
2)
2) Isosceles
1) Equilateral
Classify each triangle as Acute, Right or Obtuse

3)
4)
95 º
3) Obtuse
4) Right
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Quadrilaterals and other polygons
(rectangle, square, rhombus, parallelogram,
trapezoid)
A polygon is a plane shape with straight sides.
But the sides have to be straight, and it has to
be 2-dimensional.
A quadrilateral is a 4-sided polygon, just like a
triangle is a 3-sided polygon, a pentagon is a
5-sided polygon, and so on.
There are many different kinds of quadrilaterals,
but all have several things in common all of
them have four sides, are coplanar, have two
diagonals, and the sum of their four interior
angles equals 360 degrees.
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Types of Quadrilaterals
The Square A Square is a four-sided shape which
has all the sides equal and where every angle is
a right angle (i.e. 90).
Also opposite sides of a square are parallel.
A square also fits the definition of a rectangle
(all angles are 90), and a rhombus (all sides
are equal length).
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The Parallelogram Opposite sides are parallel
and equal in length, and opposite angles are
equal (angles "a" are the same, and angles "b"
are the same).
NOTE Squares, Rectangles and Rhombuses are all
Parallelograms!
The Trapezoid (or Trapezium) A trapezoid has
one pair of opposite sides parallel.
A trapezoid is not a parallelogram because only
one pair of sides is parallel.
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Classify each quadrilaterals as
rectangle, square, rhombus, parallelogram,
trapezoid
4)
5)
1) rectangle
2) trapezoid
6)
7)
4) parallelogram
3) square
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Congruence
Two polygons are congruent if they are the same
size and shape that is, if their corresponding
angles and sides are equal.
If one shape can become another using Turns,
Flips and/or Slides, then the two shapes are
called Congruent
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Congruent Angles
Congruent Angles have the same angle in degrees.
The angles don't have to point in the same
direction.
They don't have to be on similar sized lines.
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Congruence of triangles
A triangle has three sides and three angles.
If two triangles are congruent, then the sides
and angles that match are called corresponding
parts.
Let's look at the corresponding parts of
triangles ABC and DFE.

• Angle A corresponds to angle D.

• Angle B corresponds to angle F.

• Angle C corresponds to angle E.

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• Side AB corresponds to side DF.

• Side BC corresponds to side FE.

• Side CA corresponds to side ED.

Congruent figures are named in the order of their
corresponding parts. Here, we say "triangle ABC
is congruent to triangle DFE," because vertex A
corresponds to vertex D, vertex B corresponds to
vertex F, and vertex C corresponds to vertex E.
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Now you try!
Write whether these figures are congruent.
1)
2)
1) congruent
2) Not congruent
3)
4)
4) congruent
3) Not congruent
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BREAK
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GAME
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http//www.thekidzpage.com/onlinejigsawpuzzles/kid
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other.html
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Reflections, rotations and translations
If one shape can become another using Turns,
Flips and/or Slides, then the two shapes are
called Congruent
The three main Transformations are

• Reflection Flip!
• Rotation Turn!
• Translation Slide!

After any of those transformations (turn, flip or
slide), the shape still has the same size, area,
angles and line lengths.
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Reflection
A reflection over a line, is a transformation in
which each point of the original figure
(pre-image) has an image that is the same
distance from the line of reflection as the
original point but is on the opposite side.
The central line is called the Mirror Line, and
it doesn't matter what direction the mirror line
goes, the reflected image is always the same
size, it just faces the other way.
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Rotation
When we "rotate" an object round a point.
We can notice that
The distance from the center to any point on the
shape stays the same! and Every point makes a
circle around the center.!
"Rotation" means turning around a center.
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Rotation
A rotation is a transformation, that moves every
point around a fixed point (usually the origin).
A rotation creates a figure that is congruent to
the original figure and preserves distance and
orientation .
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Translation
In Geometry, "Translation" simply means Moving ..
without rotating, resizing or anything else, just
moving.
Every point of the shape must move
the same distance in the same
direction.
A translation is a transformation that slides
every point of a figure the same distance in the
same direction.
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Now you try!
Write Reflection, Rotation or Translation to
describe how the figure was moved
1)
2)
1) Translation
2) Reflection
3)
3) Rotation
4)
5)
4) Rotation
5) Translation, Reflection
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Similarity and Symmetry
Similar Two shapes are Similar if the only
difference is size.
If one shape can become another using Resizing,
then the shapes are Similar.
Example
When two shapes are similar, then

• corresponding angles are equal, and
• the lines are in proportion.

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Sometimes it can be hard to see if two shapes are
Similar, because you may need to turn, flip or
slide one shape as well as resizing it.
Resized and Reflected
Resized and Rotated
Resized
These shapes are all Similar.
If one shape can become another using Resizing,
then the shapes are Similar.
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Fold this picture in half. The two parts match
exactly. This picture has symmetry.
Line of symmetry
Symmetry When a picture or figure has symmetry,
it can be folded in half so that the two parts
match exactly.
Where you fold the shape, or the fold line, is
called the line of symmetry.
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Line Symmetry
A figure has line symmetry if it can be folded in
half so that the two halves match exactly i.e.
one halfof it is the mirror image of the other
half.
Line symmetry is also called bilateral symmetry.
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Figures can have any number of lines of symmetry,
from no lines of symmetry to an infinite, or
unlimited, number of lines of symmetry.
No lines of symmetry
One line of symmetry
Two lines of symmetry
Infinite lines of symmetry
The Line Symmetry is sometimes called Reflection
Symmetry or Mirror Symmetry.
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Rotational Symmetry
Rotational Symmetry A figure has rotational
symmetry if it can be rotated about a point less
than a full turn to make the figure look the same
as it did before the rotation.
3-Quarter turn
Quarter turn
Half turn
With rotational Symmetry, the shape or image can
be rotated clockwise or counterclockwise 180and
it still looks the same.
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Point Symmetry
Point Symmetry is when every part has a matching
part.
the same distance from the central point
but in the opposite direction.
Point Symmetry is sometimes called Origin
Symmetry, because the "Origin" is the central
point about which the shape is symmetrical.
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Now you try!
Write whether of figures are similar or not
1)
2)
1) similar
2) not similar
Is the dotted line a line of symmetry
3)
4)
3) Yes
4) No
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Circles and circumference (compass)
Circle A circle is a shape with all points that
are same distance from the center.
Radius
The circle is named circle O since the center is
at point O.
Radius The radius is a line segment that begins
from the centre and touches any point on the
circle.
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Diameter The distance across a circle through
the center is called the diameter.
The Diameter is equal to twice the radius.
Diameter 2 Radius
Circumference The distance around a circle is
called the circumference.
The circumference of a circle is also called the
perimeter of the circle.
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Lines in a Circle
The name of a line in a circle depends on its
position in the circle.
secant
chord
tangent
A secant is a line that passes through any two
points on a circle.
A chord is a line that joins two points on the
circumference of a circle.
A tangent is a line that touches the circle at
only one point.
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Parts of a Circle
An arc is a part of the circumference. Here, AB
is the arc.
A sector is the part of a circle between two
radii. Here, AOB is the sector .
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Parts of a Circle
A segment is the part of a circle that is between
a chord and the circumference.
A semicircle is a half of a circle.
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Circumference
The Circumference is the distance around the edge
of the circle.
It is exactly Pi (the symbol is p) times the
Diameter, so
Circumference p Diameter
Since the Diameter is equal to twice the radius.
So this is also true
Circumference 2 p Radius
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The radius of a circle is 2 inches. What is the
circumference?
The radius of a circle 2 inches
We know that,
Circumference 2 p Radius
Circumference 2 x p x 2 2 x
3.14 x 2 12.56
Replace radius with 2.
Replace p with 3.14.
The circumference of a circle 12.56 inches
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Now you try!
The radii of the circle are given. Find is the
diameter?
1) 44 cm
2) 70 ft
2) 35 ft
1) 22 cm
The radii of the circle are given. Find is the
circumference?
4) 19 ft
3) 23 cm
3) 144.44 cm
4) 119.32 ft
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A 2-dimentional figure is a shape with length and
width.
It can be open or closed.
Closed figures These are those figures that
start and end at the same point.
Open figures These are those figures that do not
start and end at the same point.
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A polygon is a closed 2-dimentional figures with
straight sides. They are made of straight lines,
and the shape is "closed.
Polygon (straight sides)
Not a Polygon (has a curve)
Not a Polygon (open, not closed)
A circle is a closed figure, but it does not have
straight sides. A circle is not a polygon.
A polygon can be grouped by the number of sides
they have.
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3-dimensional figure
A 3-dimentional figure is a figure with length,
width and height.
You can describe a 3- dimensional figure by its
parts.
vertex
A face is a flat side. A base is a face on which
the figure sits. An edge is where two faces
meet. A vertex is where 3 or more faces meet.
edge
face
base
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Examples of 3-dimensional figure.
Cube
Cylinder
A cube has 6 faces, 12 edges, 8 vertices
2 circular bases
Rectangular pyramid
Triangular pyramid
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Make a Net
You can make a net for any solid figure.
Net for a square pyramid.
Cube
Triangular Prism
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Volume of a Cuboids
Cuboids are a 3-dimensional shape.
It has 3 different measurements.
The volume of he cube is found using the formula
Volume  Height  Width  Depth
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The dimensions of a Rectangular prism is
given. Find its volume.
Height 6 cm Width 4 cm Depth 7 cm
We know that the volume of he cube is found using
the formula Volume  Height  Width  Depth
 6  4  7
 168
The volume of a Rectangular prism 168 cm2.
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Identify 3-dimentional figure. Tell how many
faces, edges and vertices it has
1)
2)
1) 4 faces, 6 edges, 4 vertices
2) 6 faces, 12 edges, 8 vertices
3)
4)
3) 1 circular base
4) no face, edge or vertices
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You have done a nice job. See you in the next
session.