Definition

1. Two figures are congruent if they both have the same shape and size. 
2. If two figures are congruent, then
(a) their corresponding angles are equal, and 
(b) their corresponding sides are equal. 
3. When naming congruent triangles, make sure that the vertices of one figure correspond to the vertices of the other figure. 

The symbol "≅" means "is congruent to".

Each of the following sets of conditions is sufficient to prove that two triangles are congruent:

Congruence and Transformation

1. To decide whether two figures are congruent, we can position one on top of the other by making three different moves:
(a) A slide is a translation.
(b) A turn is a rotation.
(c) A flip is a reflection. 
2. When we move a figure by sliding, turning and flipping, we make a transformation of the figure.
3. Under a transformation, an object is transformed onto its image. 
4. A figure and its image are congruent under a translation, a rotation and a reflection.

A. Translation
5. In a translation, every point in the figure slides the same distance and the same direction. 
6. We use a slide arrow (also called a translation vector) to show the direction and distance of the movement. 
In general, a figure is congruent to its image under a translation. If the figure we are moving is labelled ABC, then we label the transformation image with the same letters and a prime sign, i.e. A’B’C’. This is read as "A prime B prime C prime". 

B. Rotation
7. In geometry, rotating a figure means turning the figure at a point. This point is called the centre of rotation. The figure is rotated in a clockwise or anticlockwise direction through the same angle called the angle of rotation. 
In general, a figure is congruent to its image under a rotation. 

C. Reflection 
8. In geometry, a reflection is a transformation in which a figure is flipped over a line called the line of reflection.
In general, a figure is congruent to its image under a reflection. 

We can use congruent shapes to form tessellations under different transformations. In a tessellation, a shape is repeated to cover a flat surface. The shape must fit together so that there are no overlapping shapes or gaps between the shapes. 

Worked Example

Here is a worked example:

Definition

Two figures are similar if they have exactly the same shape, although they may or may not have the same size. If they have exactly the same shape and size, then they are congruent. (Congruence is a special case of similarity). 

In general, if two figures are similar, then
(a) all their corresponding angles are equal, and
(b) the ratios of all their corresponding sides are equal (i.e. the corresponding sides are proportional). 
Since there is no standard notation for similarity, we just write △ABC and △PQR are similar. 

Each of the following sets of conditions is sufficient to prove that two triangles are similar:

Proving Similarity

Conditions for two triangles to be similar:
1. All the corresponding sides are equal, OR
2. the ratios of the corresponding sides are equal. 

Conditions for two polygons with four or more sides to be similar:
1. All the corresponding sides are equal, AND
2. the ratios of the corresponding sides are equal. 

Similarity and Enlargement

1. An enlargement is a transformation that changes the size of a figure (enlarged/reduced) without changing its shape. 

2. An enlargement is determined by the centre of enlargement and a scale factor. 

3. The scale factor of an enlargement is the common ratio between pairs of corresponding sides of the image and the original figure. 
Scale factor = Length of a side of the image / Length of the corresponding side of the original figure

4. Note:
(a) If the scale factor of an enlargement is greater than 1, the image is enlarged. 
(b) If the scale factor of an enlargement is between 0 and 1, the image is reduced. 
(c) If the scale factor of an enlargement is equal to 1, the image is congruent to the original figure.
(d) An enlargement with a scale factor between 0 and 1 will give a reduced image BUT it is still called an enlargement in Mathematics. 

5. The centre of an enlargement need not be outside the original figure. It can be inside the figure, on a side or at a vertex of the original figure. 
The following video is an example of how enlargement can be done:

Similarity and Scale Drawing

1. A scale drawing is an enlarged or reduced drawing of actual objects. We use scale drawings to show objects that are too large or too small to be shown in their actual sizes. 
E.g. Maps and floor plans are examples of scale drawings. 
2. The scale factor of a scale drawing is the ratio of a length of the drawing to the corresponding length of the actual object. We can also call the scale factor of a scale drawing as the scale. 
3. The scale of a drawing is ususally written as 1 : n. It means that 1 unit length on the scale drawing represents an actual length of n units. 

The following image shows a worked example of how this is used in questions:

Worked Example

View the following worked example: