Link Search Menu Expand Document

Year 8 Maths Chapter 6 - Angle relationships and properties of geometrical figures 1

probably the stupidest section out of this entire website (so much sabotage and silly stuff)… lol good luck

You will want to refer to Year 7 Maths Chapter 4 for basic knowledge of angle relationships. This section is more advanced and omits points, rays, types of angles and other basic concepts.

Table of Contents


  • Angles sharing a vertex and an arm are called adjacent angles.
  • Two angles in a right angle are adjacent complementary angles. The first angle is the complement of the second angle.
  • You can have three or more angles in a right angle. They are not complementary.
  • Two angles on a straight line are adjacent supplementary angles. The first angle is the supplement of the second angle.
  • You can have three or more angles on a straight line. They are not supplementary.
  • Angles at a point and angles in a revolution add up to 360°.
  • Where two straight lines meet, they form two pairs of vertically opposite angles. Vertically opposite angles are equal.
  • If one of the four angles in vertically opposite is a right angle, then all four angles are right angles.

Things you should see (all images are respective to the above bullet points (going down the list, starting from the left [the two images of vertically opposite angles count as one column]):


A transversal is a line cutting two or more lines.

When a transversal crosses two or more lines, pairs of angles can be:

  • Corresponding
  • Alternate
  • Co-interior
  • Vertically opposite
  • Angles on a straight line

Corresponding, alternate and co-interior angles, respectively:

Lines are parallel if they will never intersect. They are marked with indented arrows. To mark something as parallel, you use a   sign.
Let AB be parallel to XY
∴ AB || XY
∵ I said so

You read this as:

Let AB be parallel to XY
Therefore, AB is parallel to XY
Because, I said so

Or something along those lines (hah, get it?).

If two parallel lines are cut by a transversal…

Parallel lines cut by transversal

… then corresponding and alternate angles are equal, while cointerior angles are equal to 180° (supplementary).

Fun fact! One way to remember Alternate angles is that they make a Z or a S. Salternate, Zalternate. This is sometimes not recommended by maths teachers are they get confusing.


A polygon is a type of shape where the number of interior angles equals the number of sides. Polygons can be convex or non-convex:

  • Convex polygons all have vertices pointing outside (all exterior angles are reflex angles)
  • Non-convex polygons have some vertices pointing inwards (at least one exterior angles is not a reflex angle)

A regular polygon has sides of equal length and equal interior angles. For a list of polygons, see List of n-gon names.

A triangle has:

  • 3 sides
  • 3 vertices (usually labelled as A, B, and C)
  • 1 vertex
  • 3 interior angles (usually labelled ∠ABC, ∠BAC, ∠ACB)

Triangles can be classified by interior angles:

  • Acute-angled triangles are where all angles are acute
  • Right-angled triangles are where there is one right angle
  • Obtuse-angled triangles are where there is one obtuse angle

Triangles can also be classified by their side lengths

  • Scalene triangles are where all sides are different lengths
  • Isosceles triangles are when there are two sides that are the same length
  • Equilateral triangles are when all of the sides are of equal length

For these triangles, they have unique properties:

  • Isosceles triangles have the same base angle
  • Equilateral triangles have angles that are all 60°

It is important to note that to designate sides as equal you would use a dash, similar to how you would mark parallel lines as we noted earlier.

The angle sum of a triangle is 180°.

Angle sum of n-gon

Dr Nguyen (and Osmond Lin’s dead rabbit) wanted to find the interior angle size of a polygon, but he couldn’t remember its angle sum! Dr Nguyen (and Osmond Lin’s dead rabbit) is confuddled, but by using the below steps, he can find his interior angle size of a polygon!

Let n be the number of sides of said polygon.

180(n - 2) = angle sum of n-gon

For example:

Where n-gon has 5 sides (pentagon).

180(5 - 2) = angle sum of pentagon
	     = 540°

You can therefore find the size of a single interior angle from this equation

Where the angle sum of n-gon (pentagon) is 540°

540/n = single interior angle of n-gon
540/5 = 108°

Angle sums

The exterior angle theorem dictates that the exterior angle of a triangle is equal to the sum of the two opposite interior angles.

For example:

Exterior angle theorem

The angle sum of a triangle can be used to prove other theorems, for example, one which relates to the angle in a semicircle. Although this isn’t apart of the main textbook curriculum, it is located in the Enrichment section of the textbook.

Suppose a triangle, △ABC, where BC is equal to the diameter of the semicircle, then ∠C is equal to 90°.

As you may recall from Chapter 3, quadrilaterals are four-sided polygons. They include kites, trapeziums (also known as trapezoids in North America), parallelograms, rhombus, rectangles, and squares.

Every quadrilateral have two diagonals, and in some quadrilaterals the diagonals bisect each other (cut each other in half).

Quadrilaterals can be convex or non-convex.

  • Convex quadrilaterals all have vertices pointing outside (all exterior angles are reflex angles)
  • Non-convex quadrilaterals have some vertices pointing inwards (one exterior angles is not a reflex angle)

The angle sum of a quadrilateral is 360°.

Quadrilaterals with parallel sides contain two pairs of co-interior angles.

In line and rotational symmetry, the line of symmetry divides a shape into equal parts (mirrored).

The order of rotational symmetry refers to the number of times a figure “coincides with its original position” in turning through one full rotation, according to the textbook.

A better way to put this:

In one full rotation (360° rotation) how many times does the position of the figure match the original position?

For an equilateral triangle, it is the same at the beginning of the rotation, a third of the way through, and another third of the way through, before it returns. If a figure has no rotational symmetry, its order of rotational symmetry is 1.

A polyhedron has the following properties::

  • closed solid
  • flat surfaces
  • vertices and edges
  • named after the number of faces

Euler’s formula dictates:

E = F + V - 2

Where F = faces, V = vertices and E = edges.

Prisms are polyhedra. They have the following properties:

  • two identical/congruent ends, + an identical cross section to the ends
  • other faces are parallelograms
  • right prisms have all other faces as rectangles
  • named after number of parallelograms that make up the solid

Pyramids are polyhedra with a base face, and all other faces meeting at a vertex point; the apex. They are named by the shape of the base.

Solids with curved surfaces include cylinders, spheres and cones.

Cubes (hexahedron) have 6 square, congruent faces. Rectangular prisms are also known as cuboids. Thanks America™.

Regular polygons include:

List of n-gon names

  1. Henagon
  2. Digon
  3. Triangle
  4. Quadrilateral
  5. Pentagon
  6. Hexagon
  7. Septagon
  8. Octagon
  9. Nonagon
  10. Decagon
  11. Undecagon
  12. Dodecagon
  13. Tridecagon
  14. Tetradecagon
  15. Pentadecagon
  16. Hexadecagon
  17. Heptadecagon
  18. Octadecagon
  19. Enneadecagon
  20. Icosagon
  21. Icosihenagon
  22. Icosidigon
  23. Icostrigon
  24. Icositetragon
  25. Icosipentagon
  26. Icosiexagon
  27. Icosiheptagon
  28. Isocioctagon
  29. Icosienneagon
  30. Triacontagon
  31. Triacontahenagon
  32. Triacontadigon
  33. Triacontatrigon
  34. Triacontatetragon
  35. Triacontapentagon
  36. Triacontahexagon
  37. Triacontaheptagon
  38. Triacontaoctagon
  39. Triacontaenneagon
  40. Tetracontagon
  41. Tetracontahenagon
  42. Tetracontadigon
  43. Tetracontatrigon
  44. Tetracontatetragon
  45. Tetracontapentagon
  46. Tetracontahexagon
  47. Tetracontaheptagon
  48. Tetracontaoctagon
  49. Tetracontaenneagon
  50. Pentacontagon
  51. Pentacontahenagon
  52. Pentacontadigon
  53. Pentacontatrigon
  54. Pentacontatetragon
  55. Pentacontapentagon
  56. Pentacontahexagon
  57. Pentacontaheptagon
  58. Pentacontaoctagon
  59. Pentacontaenneagon
  60. Hexacontagon
  61. Hexacontahenagon
  62. Hexacontadigon
  63. Hexacontatrigon
  64. Hexacontatetragon
  65. Hexacontapentagon
  66. Hexacontahexagon
  67. Hexacontaheptagon
  68. Hexacontatoctagon
  69. Hexacontaennagon
  70. Heptacontagon
  71. Heptacontahenagon
  72. Heptacontadigon
  73. Heptacontatrigon
  74. Heptacontatetragon
  75. Heptacontapentagon
  76. Heptacontahexagon
  77. Heptacontaheptagon
  78. Heptacontaoctagon
  79. Heptacontaenneagon
  80. Octacontagon
  81. Octacontahenagon
  82. Octacontadigon
  83. Octacontatrigon
  84. Octacontatetragon
  85. Octacontapentagon
  86. Octacontahexagon
  87. Octacontaheptagon
  88. Octacontaoctagon
  89. Octacontaennagon
  90. Enneacontagon
  91. Enneacontahenagon
  92. Enneacontadigon
  93. Enneacontatrigon
  94. Enneacontatetragon
  95. Enneacontapentagon
  96. Enneacontahexagon
  97. Enneacontaheptagon
  98. Enneacontaoctagon
  99. Enneacontaenneagon
  100. Hectogon

and of course going up in hundreds…

200. Dihectagon
300. Trihectagon
400. Tetrahectagon
500. Pentahectagon
600. Hexahectagon
700. Heptahectagon
800. Octahectagon
900. Enneahectagon
1000. Chiliagon
2000. Dischiliagon
3000. Trischiliagon
4000. Tetrakischiliagon
5000. Pentakischiliagon
6000. Hexakischiliagon
7000. Heptakischiliagon
8000. Octakichiliagon
9000. Enakichiliagon
10000. Myriagon
1000000. Megagon
∞. Apeirogon

I would just like to reinforce that I typed all of these using my fingers. The only thing copy and pasted was the infinity symbol.

Brendan’s Note: To type infinity, simply type alt-5. ∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞!


Copyright © 2017-2020 aidswidjaja and other contributors. CC BY-SA 4.0 Australia unless otherwise stated.