Congruence

Rules for congruence in triangles for GCSE maths

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Congruence

Two shapes are congruent if they are identical in shape and size

 

There are four rules that are used to prove that a pair of triangles are congruent:

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All three pairs of sides are equal (SSS)

Two triangles, left-side and right-side of both triangles are equal in length. Bottom lengths is of both triangles are equal in length  (http://www.bbc.co.uk/schools/gcsebitesize/maths/images/figure_11.gif)

When all three pairs of sides are equal, the triangles are congruent as both triangles have all sides of the same length.

This rule is written as 'SSS' for 'side side side'

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Right angle, Hypotenuse and Side are equal

image: two triangles, bottom corners of both triangles equal 90 degrees, right-side lengths of both triangle equal to each other. Left-side lengths of both triangles equal to each other.  (http://www.bbc.co.uk/schools/gcsebitesize/maths/images/figure_14.gif)

When both triangles are right angled and the lengths of the hypotenuse and the other side are equal to each other, the triangles are congruent.

e.g. when h1=h2 and s1=s2 and both are right angled triangles

This rule is written as 'RHS' for 'right angle hypotenuse side'

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Two Sides are equal and the corresponding Angles a

image: Two triangles, the right-side lengths of both triangles are equal to each other. The left-sides of both triangles are equal to each other. The top corner angle of both triangles equal to 80 degrees. (http://www.bbc.co.uk/schools/gcsebitesize/maths/images/figure_12.gif)

When  two pairs of corresponding sides are equal and the angles between the pairs of sides are also equal, then the triangles are congruent

This rule is written as 'SAS' for 'side angle side' 

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Two pairs of Angles and a pair of Sides are equal

image: Two triangles, left-side of both triangle are equal to each other. Top corners of both triangles both equal 105 degrees. (http://www.bbc.co.uk/schools/gcsebitesize/maths/images/figure_13.gif)

When two pairs of angles and a pair of corresponding sides are equal, the triangles are congruent.

This rule is written as 'AAS' for 'angle angle side'

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