CONFIDENTIAL 1

Geometry

Grade 4

CONFIDENTIAL 2

Let’s warm up :

The dimensions of Rectangular prisms are given.Find their volume.

1) Height = 5 cmWidth = 5 cmDepth = 5 cm

3) Height = 8 cmWidth = 10

cmDepth = 9 cm

2) Height = 4 cmWidth = 3 cmDepth = 6 cm

4) Height = 7 cmWidth = 7 cmDepth = 7 cm

CONFIDENTIAL 3

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.

Points, lines, segments, rays

CONFIDENTIAL 4

Point

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".

A point is a location in space.

X

CONFIDENTIAL 5

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.

D E

DE

Line

CONFIDENTIAL 6

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.

XYX Y

Line segment

CONFIDENTIAL 7

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.

Ray

A ray is named starting with its end point first and then any other point on the ray second.

O POP

This ray is called ray OP.

CONFIDENTIAL 8

Using the graphic figure:

1) Name a line.

2) Name a line segment with U as an end point.

3) Name a ray with V as an end point.

4) Name a line segment with X as an end point.

U

V

W

Y

X

Z

T

CONFIDENTIAL 9

Perpendicular, parallel & intersecting lines

Lines are parallel if they are always the same distance apart (called "equidistant"), and will

never meet.

They will travel to infinity in either direction and never intersect.

Lines m and l are parallel lines.

l

m

CONFIDENTIAL 10

Intersecting lines

Lines can only intersect at one point and only one point.

x

y

Two or more lines that meet at a point are called intersecting lines. That point would

be on each of these lines.

In the Figure, lines x and y are intersecting lines and intersect at point Q.

Q

CONFIDENTIAL 11

s

t

If the line segments meet or cross each other to form square corners, they are perpendicular to

each other.

Perpendicular lines.

The little box drawn in the corner, means "at right angles“.

right angles

Perpendicular lines intersect at a point and form 4 right angles.

CONFIDENTIAL 12

Symbol

Meaning Example In Words

|| parallelEF||GH

The line EF is parallel to line GH

perpendicular AB CD

The line AB is perpendicular to line

CD

Right angle

is 90°

A right angle is 90 degrees

Symbols in Geometry

Here are the some geometrical symbols:

CONFIDENTIAL 13

Classify each pair of lines as parallel, intersecting, or perpendicular.

4

1 2

3

Now you try!

CONFIDENTIAL 14

What Is an Angle?

An angle is a combination of two rays with a common endpoint.

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 .

OA

B

angle AOB

vertex

arm

Angles (right, acute, obtuse) protractor

CONFIDENTIAL 15

Angles On a Straight Line

If we know one angle is 45°, what is angle “x" ?

45°x

Angle x will be 180° − 45° = 135°

This method can be used to find angles on one side of a straight line.

CONFIDENTIAL 16

Angles Around a Point

40°

110°

60°150°

Angles around a point will always add up to 360 degrees.

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.

CONFIDENTIAL 17

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.

CONFIDENTIAL 18

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°.

CONFIDENTIAL 19

Find the Complement of the following:

33°

59°1 2?

?

Find the Supplement of the following:

120°?

3

135°?

4

Now you try!

CONFIDENTIAL 20

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.

The three angles always add to 180°.

vertex

arm

B

A

Ccb

a

angle ACBOr

angle c

CONFIDENTIAL 21

Interior Angle: An Interior Angle is an angle inside a shape.

Interior Angle

Exterior Angle

45°135°

Exterior Angle: The Exterior Angle is the angle between any side of a shape, and a line extended from

the next side.

If you add up the Interior Angle and Exterior Angle you get a straight line, 180°.

CONFIDENTIAL 22

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°).

B

A

Cc

ba

In the acute triangle shown above, a, b and c are all acute angles.

Triangle Classification

CONFIDENTIAL 23

Right Triangle: A right triangle is a triangle with a right angle (i.e. 90°).

hypotenuse

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.

leg

leg

b°90°

a°

right angle

CONFIDENTIAL 24

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.

>90°

a°

b°

CONFIDENTIAL 25

The triangle classification is summarized as follows:

There are three special names given to triangles that tell how many sides (or angles)

are equal.

Equilateral Triangle: An equilateral triangle has all three sides equal in length. Its three

angles are also equal and they are each 60º.

a

aa

60º 60º

60º

Types of Triangles

CONFIDENTIAL 26

Isosceles Triangle: An isosceles triangle has two sides of equal length. The angles opposite the equal sides are also equal.

a a

xº xº

Scalene Triangle: A scalene triangle has no sides of equal length. Its angles are also all

different in size.

a

b

c

xº

yº

zº

CONFIDENTIAL 27

Classify each triangle as Equilateral, Isosceles or Scalene :

2cm

2cm2cm 4 cm 4 cm

1) 2)

Classify each triangle as Acute, Right or Obtuse :

30º

90º

60º

3) 4)95 º

Now you try!

CONFIDENTIAL 28

A polygon is a plane shape with straight sides.

Quadrilaterals and other polygons (rectangle,square, rhombus, parallelogram, trapezoid)

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.

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.

CONFIDENTIAL 29

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).

CONFIDENTIAL 30

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).

a

b a

b

The Trapezoid (or Trapezium): A trapezoid has one pair of opposite sides parallel.

NOTE: Squares, Rectangles and Rhombuses are all Parallelograms!

A trapezoid is not a parallelogram because only one pair of sides is parallel.

CONFIDENTIAL 31

Classify each quadrilaterals as rectangle,square, rhombus, parallelogram, trapezoid:

a

b

a

b

1) 2)

3) 4)

Now you try!

CONFIDENTIAL 32

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:

b°

90°

a°

b°

90°

a°

CONFIDENTIAL 33

Congruent Angles

Congruent Angles have the same angle in degrees.

135°135°

The angles don't have to point in the same direction.

They don't have to be on similar sized lines.

CONFIDENTIAL 34

Congruence of triangles

If two triangles are congruent, then the sides and angles that match are called corresponding

parts. A

B C

Let's look at the corresponding parts of triangles ABC and DFE.

D

F

E

A triangle has three sides and three angles.

Angle A corresponds to angle D. Angle B corresponds to angle F. Angle C corresponds to angle E.

CONFIDENTIAL 35

A

B C D

F

E

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.

CONFIDENTIAL 36

Write whether these figures are congruent.

1) 2)

3) 4)

Now you try!

CONFIDENTIAL 37

BREAK

CONFIDENTIAL 38

Title: Format: WebEx Web BrowserDouble-click to edit

CONFIDENTIAL 39

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.

CONFIDENTIAL 40

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.

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.

Reflection

CONFIDENTIAL 41

Rotation

When we "rotate" an object round a point.

The distance from the center to any point on the shape stays the same! and

Every point makes a circle around the center.!

We can notice that

"Rotation" means turning around a center.

CONFIDENTIAL 42

A rotation is a transformation, that moves every point around a fixed point (usually the origin).

Rotation

A rotation creates a figure that is congruent to the original figure and preserves distance and

orientation .

CONFIDENTIAL 43

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.

CONFIDENTIAL 44

Write Reflection, Rotation or Translation to describe how the figure was moved:

1) 2)

3)

4)5)

Now you try!

CONFIDENTIAL 45

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.

CONFIDENTIAL 46

Resized Resized

and Reflected

Resized and

Rotated

These shapes are all Similar.

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.

If one shape can become another using Resizing, then the shapes are Similar.

CONFIDENTIAL 47

Fold this picture in half. The two parts match exactly. This picture has

“symmetry.”

Where you fold the shape, or the fold line, is called the line of 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.

CONFIDENTIAL 48

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.

CONFIDENTIAL 49

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.

CONFIDENTIAL 50

Rotational Symmetry

With rotational Symmetry, the shape or image can be rotated clockwise or counterclockwise 180°and it still

looks the same.

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.

Quarter turn

Half turn

3-Quarter turn

CONFIDENTIAL 51

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.

CONFIDENTIAL 52

Write whether of figures are similar or not:

1) 2)

Is the dotted line a line of symmetry:

3) 4)

Now you try!

CONFIDENTIAL 53

Circles and circumference (compass)

Circle: A circle is a shape with all points that are same distance from the center.

O

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.

Radius

CONFIDENTIAL 54

Diameter: The distance across a circle through the center is called the diameter.

The Diameter is equal to twice the radius.

O

Diameter

Circumference: The distance around a circle is called the circumference.

The circumference of a circle is also called the perimeter of the circle.

Diameter = 2 × Radius

CONFIDENTIAL 55

Lines in a Circle

The name of a line in a circle depends on its position in the circle.

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.

secant

chord

tangent

CONFIDENTIAL 56

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 .

AB

arc AB

O

sector AOB

CONFIDENTIAL 57

Parts of a Circle

A segment is the part of a circle that is between a chord and the circumference.

segment

A semicircle is a half of a circle.

semicircle

CONFIDENTIAL 58

Circumference

The Circumference is the distance around the edge of the circle.

It is exactly Pi (the symbol is π) times the Diameter, so:

Circumference = π × Diameter

Since the Diameter is equal to twice the radius.

So this is also true:

Circumference = 2 × π × Radius

CONFIDENTIAL 59

The radius of a circle is 2 inches. What is the circumference?

We know that,

The radius of a circle = 2 inches

Circumference = 2 × π × Radius

Circumference = 2 x π x 2 = 2 x 3.14 x 2 = 12.56

The circumference of a circle = 12.56 inches

Replace radius with 2.

Replace π with 3.14.

CONFIDENTIAL 60

The radii of the circle are given. Find is the diameter?

1) 22 cm 2) 35 ft

The radii of the circle are given. Find is the circumference?

3) 23 cm 4) 19 ft

Now you try!

CONFIDENTIAL 61

A 2-dimensional 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.

CONFIDENTIAL 62

A polygon is a closed 2-dimensional figures with straight sides. They are made of straight

lines, and the shape is "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.

Polygon (straight sides)

Not a Polygon (has a curve)

Not a Polygon (open, not closed)

CONFIDENTIAL 63

3-dimensional figure

A 3-dimensional figure: is a figure with length, width and height.

You can describe a 3- dimensional figure by its parts.

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.

vertex

face

base

edge

CONFIDENTIAL 64

Examples of 3-dimensional figure.

Cube

A cube has 6 faces, 12 edges, 8 vertices

Cylinder

2 circular bases

Triangular pyramid

Rectangular pyramid

CONFIDENTIAL 65

Make a NetYou can make a net for any solid figure.

Net for a square pyramid.

CubeTriangular Prism

CONFIDENTIAL 66

Volume of a Cuboids

Cuboids are a 3-dimensional shape.

It has 3 different measurements.

Height

WidthDep

th

The volume of he cube is found using the formula

Volume = Height × Width × Depth

CONFIDENTIAL 67

The dimensions of a Rectangular prism is given.Find its volume.

Height = 6 cmWidth = 4 cmDepth = 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.

CONFIDENTIAL 68

Identify 3-dimensional figure.Tell how many faces, edges and vertices it has

1) 2)

3) 4)

Now you try!

CONFIDENTIAL 69

You have done a nice job.Remember to do the practice sheets!