Geometry Grade 4 CONFIDENTIAL

The dimensions of Rectangular prisms are given. Let’s 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 3) Height = 8 cm Width = 10 cm Depth = 9 cm 4) Height = 7 cm Width = 7 cm Depth = 7 cm CONFIDENTIAL

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. CONFIDENTIAL

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". CONFIDENTIAL

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. CONFIDENTIAL

Line segment X XY 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. CONFIDENTIAL

This ray is called ray OP. 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. CONFIDENTIAL

Using the graphic figure: V W Y X Z T 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. CONFIDENTIAL

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

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

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. CONFIDENTIAL

Symbols in Geometry || Here are the some geometrical symbols: Symbol Meaning Example In Words || parallel EF||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 CONFIDENTIAL

Now you try! Classify each pair of lines as parallel, intersecting, or perpendicular. 1 2 3 4 CONFIDENTIAL

What Is an Angle? Angles (right, acute, obtuse) protractor An angle is a combination of two rays with a common endpoint. B angle AOB vertex O 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 . CONFIDENTIAL

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. CONFIDENTIAL

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. CONFIDENTIAL

But the angles don't have to be together to Complement each other. 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

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

Find the Complement of the following: Now you try! Find the Complement of the following: 2 1 ? 59° 33° ? Find the Supplement of the following: 3 4 120° 135° ? ? CONFIDENTIAL

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. B A C c b a angle ACB Or angle c vertex arm The three angles always add to 180°. CONFIDENTIAL

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 45° If you add up the Interior Angle and Exterior Angle you get a straight line, 180°. CONFIDENTIAL

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°). B A C c b a In the acute triangle shown above, a, b and c are all acute angles. CONFIDENTIAL

It is called the hypotenuse of the triangle. 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. b° 90° a° hypotenuse leg right angle leg CONFIDENTIAL

The longest side is always opposite the obtuse angle. Obtuse Triangle: An obtuse triangle has one obtuse angle (i.e. greater than 90º). The longest side is always opposite the obtuse angle. >90° a° b° In the obtuse triangle shown above, a is the obtuse angle. CONFIDENTIAL

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. a 60º CONFIDENTIAL

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

Classify each triangle as Equilateral, Isosceles or Scalene : Now you try! Classify each triangle as Equilateral, Isosceles or Scalene : 1) 2cm 2) 4 cm Classify each triangle as Acute, Right or Obtuse : 3) 4) 30º 90º 60º 95 º CONFIDENTIAL

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. CONFIDENTIAL

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

NOTE: Squares, Rectangles and Rhombuses are all Parallelograms! 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 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. CONFIDENTIAL

Classify each quadrilaterals as rectangle, Now you try! Classify each quadrilaterals as rectangle, square, rhombus, parallelogram, trapezoid: 1) 2) a b 3) 4) CONFIDENTIAL

Congruence Two polygons are congruent if they are the same size and shape that is, if their corresponding angles and sides are equal. b° 90° a° b° 90° a° If one shape can become another using Turns, Flips and/or Slides, then the two shapes are called Congruent: CONFIDENTIAL

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

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. A B C D F E 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. CONFIDENTIAL

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

Write whether these figures are congruent. Now you try! Write whether these figures are congruent. 1) 2) 3) 4) CONFIDENTIAL

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The three main Transformations are: 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

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. CONFIDENTIAL

"Rotation" means turning around a center. 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. CONFIDENTIAL

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 . CONFIDENTIAL

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

Now you try! Write Reflection, Rotation or Translation to describe how the figure was moved: 1) 2) 3) 4) 5) CONFIDENTIAL

Similar: Two shapes are Similar if the only difference is size. 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

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. 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. CONFIDENTIAL

Fold this picture in half. The two parts match exactly 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. CONFIDENTIAL

Line Symmetry A figure has line symmetry if it can be folded in half so that the two halves match exactly i.e. one half of it is the mirror image of the other half. Line symmetry is also called bilateral symmetry. CONFIDENTIAL

Infinite lines of symmetry 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

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 180°and it still looks the same. CONFIDENTIAL

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

Write whether of figures are similar or not: Now you try! Write whether of figures are similar or not: 1) 2) Is the dotted line a line of symmetry: 3) 4) CONFIDENTIAL

Circles and circumference (compass) Circle: A circle is a shape with all points that are same distance from the center. O 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. CONFIDENTIAL

The Diameter is equal to twice the radius. Diameter: The distance across a circle through the center is called the diameter. The Diameter is equal to twice the radius. Diameter = 2 × 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. CONFIDENTIAL

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. CONFIDENTIAL

Parts of a Circle An arc is a part of the circumference. Here, AB is the arc. A B arc AB O sector AOB A sector is the part of a circle between two radii. Here, AOB is the sector . CONFIDENTIAL

Parts of a Circle A segment is the part of a circle that is between a chord and the circumference. segment semicircle A semicircle is a half of a circle. CONFIDENTIAL

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

The radius of a circle is 2 inches. What is the circumference? The radius of a circle = 2 inches We know that, Circumference = 2 × π × Radius Circumference = 2 x π x 2 = 2 x 3.14 x 2 = 12.56 Replace radius with 2. Replace π with 3.14. The circumference of a circle = 12.56 inches CONFIDENTIAL

The radii of the circle are given. Find is the diameter? Now you try! 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 CONFIDENTIAL

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

A circle is a closed figure, but it does not have straight sides. A polygon is a closed 2-dimensional 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. CONFIDENTIAL

A 3-dimensional 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 CONFIDENTIAL

Examples of 3-dimensional figure. Cube Cylinder A cube has 6 faces, 12 edges, 8 vertices 2 circular bases Rectangular pyramid Triangular pyramid CONFIDENTIAL

Net for a square pyramid. Make a Net You can make a net for any solid figure. Net for a square pyramid. Cube Triangular Prism CONFIDENTIAL

Volume of a Cuboids Cuboids are a 3-dimensional shape. It has 3 different measurements. Height Width Depth The volume of he cube is found using the formula Volume = Height × Width × Depth CONFIDENTIAL

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. CONFIDENTIAL

Identify 3-dimensional figure. Now you try! Identify 3-dimensional figure. Tell how many faces, edges and vertices it has 1) 2) 3) 4) CONFIDENTIAL

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