6.1 Triangles

Polygons

If you take some lines and join them such that the end point of the first line meets the starting point of the last line, you will get a polygon . Each line that makes up the polygon is known as a side . A polygon has interior angles. These are the angles that are inside the polygon. The number of sides of a polygon equals the number of interior angles. If a polygon has equal length sides and equal interior angles, then the polygon is called a regular polygon . Some examples of polygons are shown in [link] .

Triangles

A triangle is a three-sided polygon. There are four types of triangles: equilateral, isosceles, right-angled and scalene. The properties of these triangles are summarised in [link] .

Name	Diagram	Properties
equilateral		All three sides are equal in length (denoted by the short lines drawn through all the sides of equal length) and all three angles are equal.
isosceles		Two sides are equal in length. The angles opposite the equal sides are equal.
right-angled		This triangle has one right angle. The side opposite this angle is called the hypotenuse .
scalene (non-syllabus)		All sides and angles are different.

If the corners of a triangle are denoted A, B and C - then we talk about $▵ABC$ .

Properties of triangles

Investigation : sum of the angles in a triangle

1. Draw on a piece of paper a triangle of any size and shape
2. Cut it out and label the angles $\widehat{A}$ , $\widehat{B}$ and $\widehat{C}$ on both sides of the paper
3. Draw dotted lines as shown and cut along these lines to get three pieces of paper
4. Place them along your ruler as shown to see that $\widehat{A}+\widehat{B}+\widehat{C}={180}^{\circ }$

Congruent triangles

Label	Description	Diagram
RHS	If the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and the respective side of another triangle, then the triangles are congruent.
SSS	If three sides of a triangle are equal in length to the same sides of another triangle, then the two triangles are congruent
SAS	If two sides and the included angle of one triangle are equal to the same two sides and included angle of another triangle, then the two triangles are congruent.
AAS	If one side and two angles of one triangle are equal to the same one side and two angles of another triangle, then the two triangles are congruent.

Similar triangles

Description	Diagram
If all three pairs of corresponding angles of two triangles are equal, then the triangles are similar.
If all pairs of corresponding sides of two triangles are in proportion, then the triangles are similar.	$\frac{x}{p}=\frac{y}{q}=\frac{z}{r}$

The theorem of pythagoras

Triangles

1. Calculate the unknown variables in each of the following figures. All lengths are in mm.

   

2. State whether or not the following pairs of triangles are congruent or not. Give reasons for your answers. If there is not enough information to make adescision, say why.