Geometric Reasoning
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Transcript of Geometric Reasoning
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Geometric ReasoningGeometric Reasoning
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Types of AnglesTypes of AnglesName of Angle Picture Description
Acute angle Less than 90˚
Right angle Exactly 90˚
Obtuse angle Between 90˚ & 180˚
Straight angle Exactly 180˚
Reflex angle between 180˚ & 360˚
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PolygonsPolygons
A Polygon is a closed figure made up of straight sides. They are given special names when we know the number of sides.
A Polygon is a closed figure made up of straight sides. They are given special names when we know the number of sides.
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PolygonsPolygons
A regular polygon is one that has all its sides and angles the same. An irregular polygon does not.
Examples of regular polygons
A regular polygon is one that has all its sides and angles the same. An irregular polygon does not.
Examples of regular polygons
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Types of TrianglesTypes of TrianglesReason Picture Sides AnglesScalene Triangle
No equal sides
No equal angles
Isosceles Triangle
2 equal sides
2 equal angles
Equilateral Triangle
3 equal sides
3 equal angles (all 60˚)
Acute Triangle
All angles less than 90˚
Right Angled Triangle
One right-angle (90˚)
Obtuse Triangle
One angle greater than 90˚
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Types of Quadrilaterals
Types of Quadrilaterals
There are several quadrilaterals including Square, Rectangle, Parallelogram, Rhombus.
Quadrilaterals are a type of polygon with four sides, and four angles adding up to 360°.
There are several quadrilaterals including Square, Rectangle, Parallelogram, Rhombus.
Quadrilaterals are a type of polygon with four sides, and four angles adding up to 360°.
A pushedover square
A pushedover rectangle
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Exterior Angles of Polygons
Exterior Angles of Polygons
This is easy, they add up to 360°. Think of the opening of a camera. As it gets smaller and smaller it comes to a point. (360º)
This is easy, they add up to 360°. Think of the opening of a camera. As it gets smaller and smaller it comes to a point. (360º)
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Interior Angles of Polygons
Interior Angles of Polygons
The formula for calculating the sum of the interior angles of a regular polygon is:
(n - 2) × 180° where n is the number of sides of the polygon.
The formula for calculating the sum of the interior angles of a regular polygon is:
(n - 2) × 180° where n is the number of sides of the polygon.
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Interior angle of a regular polygon
Interior angle of a regular polygon
Example: Find the interior angle of a regular hexagon.
You know that the interior angles of a hexagon add up to 720°As a hexagon has six sides, each angle is equal to = 120°.
Example: Find the interior angle of a regular hexagon.
You know that the interior angles of a hexagon add up to 720°As a hexagon has six sides, each angle is equal to = 120°.
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BearingsBearings Bearings are special angles that give directions. They are measured clockwise from North, and are always written using three digits.
EG:
Bearings are special angles that give directions. They are measured clockwise from North, and are always written using three digits.
EG: N
0700
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ExercisesExercises
Types of angles: Exercise 9.3 All
Bearings: Exercise 9.4 All
Types of angles: Exercise 9.3 All
Bearings: Exercise 9.4 All
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Angle ReasoningAngle ReasoningReason Picture Short-hand
Angles on a straight line add to 180˚
’s on line
Vertically opposite angles are equal
vert opp ’s
Angles at a point add to 360˚
’s at pt
Angles in a triangle add to 180˚
sum of
The exterior angle of a triangle is equal to the sum of the two interior opposite angles
ext of
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The base angles of an isosceles triangle are equal
base ’s isos
Each angle in an equilateral triangle = 60˚
equilat
Complementary angles add to 90˚
32˚ is the complement of 58˚
Supplementary angles add to 180˚
70˚ is the supplement of 110˚
Reason Picture Example Short-hand
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ExercisesExercises
Lines, Points and Triangles: Exercise 9.5 All Exercise 9.6 All Exercise 9.7 All
Remember to give the right reason for your answer! i.e x = 700: supplementary angles add to 1800
Lines, Points and Triangles: Exercise 9.5 All Exercise 9.6 All Exercise 9.7 All
Remember to give the right reason for your answer! i.e x = 700: supplementary angles add to 1800
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Parallel LinesParallel LinesReason Picture Example Short-hand
Corresponding angles on parallel lines are equal
Angle A = Angle B
corresp ’s, // lines
Alternate angles on parallel lines are equal
Angle I = Angle J
Alt ’s, // lines
Co-interior angles on parallel lines are supplementary (add to 180˚)
E + F = 180˚
If E = 120˚ then F = 60˚
Co-int ’s, // lines
A
B
I
J
E F
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ExercisesExercises
Parallel Lines: Exercise 9.8 All
Parallel Lines Solving for x Exercise 9.9 All
Remember to give the right reason for your answer! i.e x = 700: Alternate angles on parallel lines are equal
Parallel Lines: Exercise 9.8 All
Parallel Lines Solving for x Exercise 9.9 All
Remember to give the right reason for your answer! i.e x = 700: Alternate angles on parallel lines are equal
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Parts of a circleParts of a circleCircumference The distance
around the circle
Radius The distance from the centre to a point on the circumference
Diameter A chord that passes through the centre
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Name Description Picture
ArcMinor arc
Major arc
A part of the circumferenceLess than half of the circumference
More than half of the circumference
Chord A line joining two points on the circumference
Segment Part of a circle bounded by an arc and a chord
Sector Part of a circle bounded by an arc and two radii
Tangent A line that touches the circumference of the circle at only one point
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ExerciseExercise
Parts of a circle: Exercise 10.1 All
Parts of a circle: Exercise 10.1 All
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Angle Properties of Circles
Angle Properties of Circles
Name Description Picture Short-hand
Radii Two radii in a circle form an isosceles triangle.
OAB is an isosceles triangle. Angle A = Angle B
isos , = radii
base ’s isos , = radii
sum isos , = radii
Angle at centre(Pg.124)
The angle at the centre is twice the angle at the circumference
e.g. B = 2 x AIf A = 550 B = 2x55 =110o
at centre
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Angle in a semi-circle
Interior angle in a semicircle is 180o and so angle at circumference is 90o
ACB = ½ x 180o = 90o
in a semi-circle
Angles on same arc
Angles extending to the circumference from the same arc are equali.e. a = b
’s on same arc
Name Description Picture Short-hand
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ExerciseExercise
Properties of a circle: Exercise 10.2 All Exercise 10.3 All Exercise 10.4 All
Remember to give the right reason for your answer! i.e x = 250: The angle at the centre is twice the angle at the circumference
Properties of a circle: Exercise 10.2 All Exercise 10.3 All Exercise 10.4 All
Remember to give the right reason for your answer! i.e x = 250: The angle at the centre is twice the angle at the circumference
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Rotational SymmetryRotational Symmetry A figure has rotational symmetry about a point if it can rotate onto itself in less then 3600.
If a shape only rotates onto itself once then it is said to not have rotational symmetry
Order of Rotational Symmetry
The order of rotational symmetry is how often a shape will rotate onto itself
Every shape will have a rotational symmetry of at least 1
A figure has rotational symmetry about a point if it can rotate onto itself in less then 3600.
If a shape only rotates onto itself once then it is said to not have rotational symmetry
Order of Rotational Symmetry
The order of rotational symmetry is how often a shape will rotate onto itself
Every shape will have a rotational symmetry of at least 1
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Line SymmetryLine Symmetry
A shape has line symmetry if it reflects or folds onto itself. The line or fold is called an axis of symmetry
Use a ruler to help you work out how many axis of symmetry a shape has
A shape has line symmetry if it reflects or folds onto itself. The line or fold is called an axis of symmetry
Use a ruler to help you work out how many axis of symmetry a shape has
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Total Order of Symmetry
Total Order of Symmetry
The Total Order of Symmetry of a shape is:
The number of Axis of Symmetryplus
The Order of Rotational Symmetry
The Total Order of Symmetry of a shape is:
The number of Axis of Symmetryplus
The Order of Rotational Symmetry