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# Exploring congruence

Shapes which can be superimposed on each other are said to be congruent. They are the result of a rotation, translation or reflection or a combination of these transformations. Congruent shapes are used extensively in design.

• Explore real life patterns such as those in tiling and patchwork.

Patterned floor tiles.
Source: Getty/Pete Ryan.

• Cut out shapes from within a design and place them on top of each other.
• Discuss the sequence of transformations required to map one shape onto another: translation, rotation and reflection.
• Use pattern blocks to construct designs based on congruence.
• Provide students with a cut-out of a specific shape which they can use to re-draw the shape in a new orientation.

For example, draw ABCD in the new orientation indicated by A’B’.

Subsequently, provide students with activities which require them to draw transformations of shapes without the aid of a real object.

Students find challenge in constructing designs which involve rotations.

• Provide objects which can be manipulated physically to assist.
• Use polar grid paper to complete the rotation and then trace the resulting shape onto square grid paper.
• Highlight the matching sides and angles in different colours to ensure that they are equal.

## Developing the congruence tests

In order to establish the minimal conditions under which triangles are congruent, explore all triangles which satisfy three given conditions.