geometric reasoning

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