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: <span data-mce-type="bookmark" style="display: inline-block; width: 0px; overflow: hidden; line-height: 0;" class="mce_SELRES_start">﻿</span> Con­struct­ing Triangles

### SAS andASA

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°

## Poly­gons

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°

OR

Inte­rior angle = 180° — Exte­rior angle

OR

Exte­rior angle = 180° — Inte­rior angle.