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
- 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…
… 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.
Polygons
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:
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
- Henagon
- Digon
- Triangle
- Quadrilateral
- Pentagon
- Hexagon
- Septagon
- Octagon
- Nonagon
- Decagon
- Undecagon
- Dodecagon
- Tridecagon
- Tetradecagon
- Pentadecagon
- Hexadecagon
- Heptadecagon
- Octadecagon
- Enneadecagon
- Icosagon
- Icosihenagon
- Icosidigon
- Icostrigon
- Icositetragon
- Icosipentagon
- Icosiexagon
- Icosiheptagon
- Isocioctagon
- Icosienneagon
- Triacontagon
- Triacontahenagon
- Triacontadigon
- Triacontatrigon
- Triacontatetragon
- Triacontapentagon
- Triacontahexagon
- Triacontaheptagon
- Triacontaoctagon
- Triacontaenneagon
- Tetracontagon
- Tetracontahenagon
- Tetracontadigon
- Tetracontatrigon
- Tetracontatetragon
- Tetracontapentagon
- Tetracontahexagon
- Tetracontaheptagon
- Tetracontaoctagon
- Tetracontaenneagon
- Pentacontagon
- Pentacontahenagon
- Pentacontadigon
- Pentacontatrigon
- Pentacontatetragon
- Pentacontapentagon
- Pentacontahexagon
- Pentacontaheptagon
- Pentacontaoctagon
- Pentacontaenneagon
- Hexacontagon
- Hexacontahenagon
- Hexacontadigon
- Hexacontatrigon
- Hexacontatetragon
- Hexacontapentagon
- Hexacontahexagon
- Hexacontaheptagon
- Hexacontatoctagon
- Hexacontaennagon
- Heptacontagon
- Heptacontahenagon
- Heptacontadigon
- Heptacontatrigon
- Heptacontatetragon
- Heptacontapentagon
- Heptacontahexagon
- Heptacontaheptagon
- Heptacontaoctagon
- Heptacontaenneagon
- Octacontagon
- Octacontahenagon
- Octacontadigon
- Octacontatrigon
- Octacontatetragon
- Octacontapentagon
- Octacontahexagon
- Octacontaheptagon
- Octacontaoctagon
- Octacontaennagon
- Enneacontagon
- Enneacontahenagon
- Enneacontadigon
- Enneacontatrigon
- Enneacontatetragon
- Enneacontapentagon
- Enneacontahexagon
- Enneacontaheptagon
- Enneacontaoctagon
- Enneacontaenneagon
- 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. ∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞!
Source: https://en.wikipedia.org/wiki/List_of_polygons#List_of_n-gons_by_Greek_numerical_prefixes