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Geometric Reasoning. Types of Angles. Polygons. A Polygon is a closed figure made up of straight sides. They are given special names when we know the number of sides. Polygons. A regular polygon is one that has all its sides and angles the same. An irregular polygon does not. - PowerPoint PPT Presentation
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Geometric ReasoningGeometric Reasoning
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˚
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.
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
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˚
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
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)
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.
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°.
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
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
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
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
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
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
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
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
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
ExerciseExercise
Parts of a circle: Exercise 10.1 All
Parts of a circle: Exercise 10.1 All
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
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
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
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
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
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
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