confidential1 geometry grade 4. confidential2 let’s warm up : the dimensions of rectangular prisms...
TRANSCRIPT
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!