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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 Figure 1.

Figure 1: Examples of polygons. They are all regular, except for the one marked *
Figure 1 (MG10C13_0221.png)

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 Table 1.

Table 1: Types of Triangles
Name Diagram Properties
equilateral
Figure 2
Figure 2 (MG10C13_023.png)
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
Figure 3
Figure 3 (MG10C13_024.png)
Two sides are equal in length. The angles opposite the equal sides are equal.
right-angled
Figure 4
Figure 4 (MG10C13_025.png)
This triangle has one right angle. The side opposite this angle is called the hypotenuse.
scalene (non-syllabus)
Figure 5
Figure 5 (MG10C13_026.png)
All sides and angles are different.

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

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 A^A^, B^B^ and C^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 A^+B^+C^=180A^+B^+C^=180

Figure 6
Figure 6 (MG10C13_027.png)
Figure 7
Figure 7 (MG10C13_028.png)

Tip:
The sum of the angles in a triangle is 180.
Figure 8: In any triangle, A+B+C=180A+B+C=180
Figure 8 (MG10C13_029.png)
Tip:
Any exterior angle of a triangle is equal to the sum of the two opposite interior angles. An exterior angle is formed by extending any one of the sides.
Figure 9: In any triangle, any exterior angle is equal to the sum of the two opposite interior angles.
Figure 9 (MG10C13_030.png)

Congruent Triangles

Table 2
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.
Figure 10
Figure 10 (MG10C13_031.png)
SSS If three sides of a triangle are equal in length to the same sides of another triangle, then the two triangles are congruent
Figure 11
Figure 11 (MG10C13_032.png)
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.
Figure 12
Figure 12 (MG10C13_033.png)
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.
Figure 13
Figure 13 (MG10C13_034.png)

Similar Triangles

Table 3
Description Diagram
If all three pairs of corresponding angles of two triangles are equal, then the triangles are similar.
Figure 14
Figure 14 (MG10C13_035.png)
If all pairs of corresponding sides of two triangles are in proportion, then the triangles are similar.
Figure 15
Figure 15 (MG10C13_036.png)
x p = y q = z r x p = y q = z r

The theorem of Pythagoras

Figure 16
Figure 16 (MG10C13_037.png)
If ABC is right-angled (B^=90B^=90) then b2=a2+c2b2=a2+c2
Converse: If b2=a2+c2b2=a2+c2, then ABC is right-angled (B^=90B^=90).

Triangles
  1. Calculate the unknown variables in each of the following figures. All lengths are in mm.
    Figure 17
    Figure 17 (MG10C13_038.png)
    Click here for the solution
  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 a descision, say why.
    Figure 18
    Figure 18 (MG10C13_039.png)
    Click here for the solution

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