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1. Good Afternoon! Today we will be learning about Review of Geometry 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 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 CONFIDENTIAL

2. 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

3. Point X Apointis 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

4. Line DE E D Aline 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

5. Line segment XY X Y Aline segmentis 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

6. 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. CONFIDENTIAL

7. X T U Y V W Z 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 CONFIDENTIAL

8. Perpendicular, parallel & intersecting lines Lines areparallelif they are always the same distance apart (called "equidistant"), and will never meet. l m Lines m and lare parallel lines. They will travel to infinity in either direction and never intersect. CONFIDENTIAL

9. 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

10. 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

11. Symbols in Geometry Here are the some geometrical symbols: CONFIDENTIAL

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

13. Angles (right, acute, obtuse) protractor What Is an Angle? 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 sidesor arms of the angle . CONFIDENTIAL

14. 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

15. 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. CONFIDENTIAL

16. 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

17. Supplementary Angles Two Angles are Supplementary if they add up to 180 degrees(a StraightAngle). 120° 60° These two angles (120° and 60°) are Supplementary Angles, because they add up to 180°. CONFIDENTIAL

18. 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° ? ? CONFIDENTIAL

19. A a B c C b 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°. CONFIDENTIAL

20. 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 Angleyou get a straight line, 180°. CONFIDENTIAL

21. A a B b c C 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. CONFIDENTIAL

22. a° 90° b° 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 CONFIDENTIAL

23. a° >90° b° 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. CONFIDENTIAL

24. a a 60º 60º 60º a 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. CONFIDENTIAL

25. a a xº xº b xº zº a c yº 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. CONFIDENTIAL

26. 4 cm 4 cm 30º 60º 90º 2cm 2cm 2cm 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 CONFIDENTIAL

27. 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

28. Types of Quadrilaterals The Square:A Square is a four-sided shape which has all the sides equal andwhere 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

29. a b a b 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. CONFIDENTIAL

30. a b a b Classify each quadrilaterals as rectangle, square, rhombus, parallelogram, trapezoid: 4) 5) 1) rectangle 2) trapezoid 6) 7) 4) parallelogram 3) square CONFIDENTIAL

31. a° a° 90° 90° b° b° 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: CONFIDENTIAL

32. 135° 135° 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. CONFIDENTIAL

33. E A D F C B 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. CONFIDENTIAL

34. A E D C B F • 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

35. Now you try! Write whether these figures are congruent. 1) 2) 1) congruent 2) Not congruent 3) 4) 4) congruent 3) Not congruent CONFIDENTIAL

36. BREAK CONFIDENTIAL

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38. 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 hasthe same size,area, anglesandline lengths. CONFIDENTIAL

39. 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 MirrorLine, 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

40. 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. CONFIDENTIAL

41. 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

42. 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

43. 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 CONFIDENTIAL

44. 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

45. 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

46. 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

47. 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 bilateralsymmetry. CONFIDENTIAL

48. 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 ReflectionSymmetry or MirrorSymmetry. CONFIDENTIAL

49. 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

50. 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