Interior and Exterior Angles of a Triangle
First a couple of rules using this triangle:
Rule 1: The sum of the angles of a triangle (x + y + z) equals 180°
Rule 2: The exterior angle (angle w, above) of a triangle is equal to the sum of the two opposite angles (x and y, above)
Note that an exterior angle lies outside the triangle and is made by extending one sides of the triangle.
This video shows these 2 rules in action:
SAS and ASA
SAS: If you know the length of 2 sides and one angle you can construct a full triangle.
ASA: If you know 2 angles and the length of one side between those angles you can complete a triangle.
Here’s a video showing the construction of a triangle based on SAS:
Here’s a video showing the construction of a triangle using ASA:
A quadrilateral is a shape formed by 4 straight lines. These are all types of quadrilateral:
The above were some special, named quadrilaterals. All quadrilaterals have certain attributes:
- Quadrilaterals are 2D shapes bounded by four straight lines.
- The diagonal – a straight line joining two opposite vertices or corners– divides the quadrilateral into two triangles.
We know that the sum of a triangle’s interior angles adds up to 180° so it follows that if quadrilaterals can be divided into 2 triangles then the sum of a quadrilateral’s interiors angles add up to 180° plus 180° = 360°
There are a number of rules about ploygons. Quite often questions will ask you to use these rules to calculate interior or exterior angles.
A polygon is a two dimensional shape bounded by straight lines, so a triangle is a polygon.
Regular polygons have sides that are the same length and all the angles in a regular polygon are the same size.
These are all regular polygons:
3 sides — Equilateral triangle
4 sides — Square
5 sides — Regular Pentagon
6 sides — Regular Hexagon
7 sides — Regular Heptagon
8 sides — Regular Octagon
The sum of the exterior angles of any polygon is 360°
In a regular polygon all the exterior angles are the same size and can be calculated using this rule:
Exterior angle of a regular polygon = 360°/number of sides
So for a regular pentagon (5 sides) each exterior angle will be:-
360°/5 = 72°
Quadrilaterals (see above) are four-sided polygons. We saw that a quadrilateral could be divided into 2 triangles by drawing a line from one vertex (corner) to the opposite vertex and that this proved that the interior angles of a quadrilateral were 360° (180° + 180°).
In a similar way any polygon can be divided into a number of triangles. A quadrilateral (4 sides) can be divided into 2 triangles, a pentagon (5 sides) can be divided into 3 triangles and hexagon can be divided into 4 triangles. Can you see the pattern or rules that’s emerging? –
The sum of the interior angles of a polygon is (number of sides — 2) x 180°
When you consider a polygon’s vortex and it’s interior and exterior angle, note that the interior and exterior angles lie on a straight line so:
Interior angle + Exterior angle = 180°
Interior angle = 180° — Exterior angle
Exterior angle = 180° — Interior angle.