Geometry

Cameron Ualesi, Nov 8, 2018

Table of Contents

  1. How to use the geogebra tools
    1. Using the angle tool
    2. Using the distance and length tool
  2. Triangles
    1. Properties of an equilateral triangle
    2. Properties of an isosceles triangle
    3. Properties of a right angled triangle
    4. Properties of a scalene triangle
  3. Quadrilaterals
    1. Properties of a rectangle
    2. Properties of a square
    3. Properties of a rhombus
    4. Properties of a parallelogram
    5. Properties of a trapezium
    6. Properties of a kite
  4. Interior and exterior angles of polygons
    1. Interior angles of polygons
    2. Exterior angles of polygons
  5. Naming Angles
    1. Types of angles
    2. Denoting angles
  6. Angle Rules
    1. Basic rules
    2. Vertically opposite angles
  7. Angle rules for parallel lines
    1. Corresponding Angles on Parallel Lines
    2. Alternate Angles on Parallel Lines
    3. Co-interior Angles on Parallel Lines
  8. Pythagoras
    1. Pythagoras
  9. Trigonometry relationships
    1. Labelling the sides of a right angle triangle
    2. Introduction to Sine
    3. Introduction to Cosine & Tangent
    4. "Soh Cah Toa"
    5. Using Sine to find a side
    6. Using Cosine to find a side
    7. Using Tangent to find a side
    8. Using sine to find an angle
    9. Using cosine to find an angle
    10. Practice using trigonometry ratios
  10. Circle Theorems
    1. Circle properties
    2. Angle at the centre rule:
    3. Angles on the same arc rule:
    4. Angle in a semi-circle rule:
    5. Angles in a cyclic quadrilateral:
    6. Tangent angle:
    7. Practice of circle theorems:

1.

How to use the geogebra tools

  • 1. Using the angle tool
  • 2. Using the distance and length tool

1.1.

Using the angle tool

When using the angle tool you must select the points in a [b]clockwise direction.[/b]
Have a go yourself.
– Cameron Ualesi

1.2.

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2.

Triangles

  • 1. Properties of an equilateral triangle
  • 2. Properties of an isosceles triangle
  • 3. Properties of a right angled triangle
  • 4. Properties of a scalene triangle

2.1.

Properties of an equilateral triangle

Equilateral triangle
Add all the interior angles and the lengths of the sides
What do you notice about the sides?
What do you notice about all the angles?
– Cameron Ualesi

2.2.

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2.3.

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2.4.

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3.

Quadrilaterals

  • 1. Properties of a rectangle
  • 2. Properties of a square
  • 3. Properties of a rhombus
  • 4. Properties of a parallelogram
  • 5. Properties of a trapezium
  • 6. Properties of a kite

3.1.

Properties of a rectangle

Add all interior angles and lengths of the sides.
What do you notice about the lengths of the sides?
What do you notice about the interior angles?
Are there any sides that are parallel?
– Cameron Ualesi

3.2.

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3.3.

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3.4.

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3.5.

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3.6.

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4.

Interior and exterior angles of polygons

  • 1. Interior angles of polygons
  • 2. Exterior angles of polygons

4.1.

Interior angles of polygons

Triangle
What is the sum of all the interior angles?
Square
What is the sum of all the interior angles?
Pentagon
What is the sum of all the interior angles?
Hexagon
What is the sum of all the interior angles?
Heptagon
What is the sum of all the interior angles?
Have a look back at the answers you have picked. Can you recognize a pattern when adding an extra side?
What do you think the pattern is?[br]
[b]Now have a ago at using this formula:[br][br]Sum of interior angles = (n-2) x 180[br][br]See if you get the same answers.[/b]
Does the formula work only for regular polygons?
What kind of polygon is this?
Add up all the angles and also use the formula.[br]Do you get the same answers?
– Cameron Ualesi

4.2.

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5.

Naming Angles

  • 1. Types of angles
  • 2. Denoting angles

5.1.

Types of angles

What type of angle is this?
Because......
What type of angle is this?
Because......
What type of angle is this?
Because......
What type of angle is this?
Because......
What type of angle is this?
Because......
– Cameron Ualesi

5.2.

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6.

Angle Rules

  • 1. Basic rules
  • 2. Vertically opposite angles

6.1.

Basic rules

Complementary angles
Complementary angles are a pair of angles that add to 90 degrees
You can see in the video that [math]\angle[/math]DBA = 68.2 and [math]\angle[/math]CBD = 21.8[br]68.2 + 21.8 = 90[br]Therefore the angles are [b]complementary[/b]
Use the angle tool to find all angles that are complementary
Which angles are complementary?[br]
Supplementary angles
Supplementary angles are a pair of angles that add to 180 degrees
You can see in the video that [math]\angle[/math]DBA = 75.96 and [math]\angle[/math]CBD =104.04[br]75.96 + 104.04 = 180[br]Therefore the angles are [b]supplementary[/b]
Which pairs of angles are supplementary?
Angles in a Triangle
All three angles inside a triangle add to 180 degrees
You can see in the video that the three interior angles are:[br][math]\angle[/math]CAB = 27.05[math]^\circ[/math][br][math]\angle[/math]ABC = 114.17[math]^\circ[/math][br][math]\angle[/math]BCA = 38.78[math]^\circ[/math][br][br]27.05 + 114.17 + 38.78 = 180
Find the missing angle [math]\angle[/math]CBA
Use what you have learnt to find [math]\angle[/math]EFC
Angles on a line add to 180 degrees
Go to this transum website to have a go:[br]https://www.transum.org/Software/SW/Starter_of_the_day/Students/AnglePoints.asp?Level=2
Angles at a point
Move any point around and watch how the angles change.[br]But what do you notice about the sum of the angles?
What is the value of [math]\angle[/math]FBE?
For more questions complete this activity:[br]https://www.transum.org/Software/SW/Starter_of_the_day/Students/AnglePoints.asp
– Simon Job, Duane Habecker, Cameron Ualesi

6.2.

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7.

Angle rules for parallel lines

  • 1. Corresponding Angles on Parallel Lines
  • 2. Alternate Angles on Parallel Lines
  • 3. Co-interior Angles on Parallel Lines

7.1.

Corresponding Angles on Parallel Lines

What do you notice about the corresponding angles of parallel lines?
What would be the value of [math]\angle[/math]AFE?
What would be the value of [math]\angle[/math]CEH
How do we know this?
If you would like more questions click this link:[br]https://www.transum.org/software/SW/Starter_of_the_day/Students/AngleParallel.asp?Level=2
– Simon Job, Cameron Ualesi

7.2.

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7.3.

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8.

Pythagoras

  • 1. Pythagoras

8.1.

Pythagoras

Drag the coloured pieces over the correct dots.
[b]1) What do you notice?[br]2) What does it tell us?[/b]
Move the red and green sliders and watch what happens to the Blue area.
What do you notice if you add the red area and the green area together?
Pythagoras Theorem
If we denote the sides a, b, and c we can say:[br]Red area = a[math]^2[/math][br]Green area = b[math]^2[/math][br]Blue area = c[math]^2[/math][br][br]We found previously that:[br]Red area + Green area = Blue area[br][br]So;[br][math]a^2+b^2=c^2[/math] and this is [b]Pythagoras theorem[/b]
What kind of triangles are the three used above?
Finding the long side
Pythagoras is used if you have [b]two[/b] sides of a [b]right angle triangle [/b]and you want to find the third side. [br][br][b]To find the longest side:[/b][br] [math]c=\sqrt{a^2+b^2}[/math][br][br][b]To find the short side:[/b][br][math]b=\sqrt{c^2-a^2}[/math][br]or[br][math]a=\sqrt{c^2-b^2}[/math]
If a = 4 and b =7,[br]what is c[math]^2[/math]
So then what would [b]c[/b] be?
Finding a short side.
If c = 6.25 and b = 5.34,[br]what is a[math]^2[/math]
So then what would [b]a[/b] be?
To practice finding the long side, complete this activity:[br]https://www.transum.org/software/SW/Starter_of_the_day/Students/Pythagoras_basics.asp?Level=1[br][br]To practice finding the short side, complete this activity:[br]https://www.transum.org/software/SW/Starter_of_the_day/Students/Pythagoras_basics.asp?Level=2
– Steve Phelps, Cameron Ualesi

9.

Trigonometry relationships

  • 1. Labelling the sides of a right angle triangle
  • 2. Introduction to Sine
  • 3. Introduction to Cosine & Tangent
  • 4. "Soh Cah Toa"
  • 5. Using Sine to find a side
  • 6. Using Cosine to find a side
  • 7. Using Tangent to find a side
  • 8. Using sine to find an angle
  • 9. Using cosine to find an angle
  • 10. Practice using trigonometry ratios

9.1.

Labelling the sides of a right angle triangle

Have a go at labeling the sides. Check if you are correct by ticking the boxes.
Have a go at labeling the sides. Check if you are correct by ticking the boxes.
– Cameron Ualesi

9.2.

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9.3.

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9.4.

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9.5.

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9.6.

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9.7.

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9.8.

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9.9.

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9.10.

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10.

Circle Theorems

  • 1. Circle properties
  • 2. Angle at the centre rule:
  • 3. Angles on the same arc rule:
  • 4. Angle in a semi-circle rule:
  • 5. Angles in a cyclic quadrilateral:
  • 6. Tangent angle:
  • 7. Practice of circle theorems:

10.1.

Circle properties

Circle properties
– Gerrit Stols

10.2.

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10.3.

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10.4.

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10.5.

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10.6.

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10.7.

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