Tests for congruency
If two triangles are congruent they must have the same shape and the same size. This means that they will fit exactly onto each other when one of them is rotated, reflected or translated.
Two triangles are congruent if any of these conditions are satisfied:
- two sides and the included angle of one triangle are equal to the two sides and the included angle of the other triangle (side, angle, side or SAS)
- the three sides of one triangle are equal to the corresponding three sides of the other triangle (side, side, side or SSS)
- two angles and a side in one triangle are equal to two angles and the corresponding side in the other triangle (angle, corresponding angle side or AAS)
- each triangle is right-angled and the hypotenuse and one side of one triangle is equal to the hypotenuse and a side of the other triangle, (right angle, hypotenuse, side or RHS). The hypotenuse is the side opposite the right angle.
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The tests for congruency: SSS; SAS; ASA are essentially the methods used to construct triangles
(a) SSS
(i) Draw the line AB of given length.
(ii) Use compasses to construct arcs AC and BC.
(iii) Draw AC and BC.
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(b) SAS
(i) Draw a line AB of given length.
(ii) Measure and draw the angle at A.
(iii) Draw the line AC of given length.
(iv) Join C to B.
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(c) ASA
(i) Draw the line AB of given length.
(ii) Measure and draw the angles at A and B.
(iii) Draw lines AC and BC.
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The basic checks for congruency are:
(a) the three sides of each triangle are equal
(b) two sides and the included angle in each triangle are equal
(c) two angles and the corresponding side in each triangle are equal
(d) for a right-angled triangle the check is ‘right angle, hypotenuse and side’ in each triangle are equal.
angle a = angle b (alternate angles)
angle a = angle c (corresponding angles)
The test involving two sides and an angle is valid only if the angle is the included angle. If the given angle is not the included angle then there are two possible solutions: the ambiguous case.
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