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Congruence and the Ambiguous Case
Two triangles which are identical or are perfect mirror images of each other are said to be
congruent. Note, they may not at first sight look identical because the two triangles may be
differently oriented. This does not matter for congruence: only corresponding sides and angles on
the two triangles have to be equal for them to be congruent. They could be rotated and moved to
either lie one top of the other or, if they are mirror images, placed face to face like your right and
left hand. There are minimum conditions for triangles to meet for them to be congruent.
The conditions for congruence of a pair of triangles are covered by the four cases:




SSS – all three corresponding pairs of sides are equal
SAS – two corresponding pairs of sides, and the corresponding pair of angles included
between them are equal
ASA – two corresponding pairs of angles and a corresponding pair of sides included between
them equal
RHS – both triangles have a right-angle, equal hypotenuses and one other pair of
corresponding sides equal
It may be wondered as to why other combinations of angles and sides do not give congruence. If
there is less than three items of agreement (two pairs of sides, or one pair of sides and one pair of
angles, say) then it can be easily seen that this gives too much latitude to define a unique triangle
and an infinite number of different pairs of triangles meeting these limited conditions can be
constructed. The combination AAA (all three pairs of corresponding angles equal) gives, not
congruent triangles, but instead, similar triangles in which one triangle is any scaled enlargement or
reduction of the other.
θB
Similar Triangles –
Case AAA
θB
θC
θC
θA
θA
There are three other missing combinations, SRS, AAS and ASS (the last combination of letters
should give you pause should you encounter the situation). The first of these (SRS) is easily
dispensed with: the combination of two pairs of sides being equal and right-angles being contained
in both triangles between these sides is just a special case of SAS. Alternatively, applying Pythagoras’
Theorem shows that the two hypotenuses (the third and longest sides) are equal so we can equally
well interpret this as a case of RHS congruence.
Bill Bavington
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The combination AAS is two pairs of corresponding angles equal and one pair of corresponding sides
equal that are not between the equal angles. Since the sum of all three interior angles of a triangle is
always 180°, two pairs of angles being equal in two triangles is sufficient to make the third pair equal
as well, giving the ASA condition for congruence. More explicitly, in triangles ABC and DEF, angle BAC
= angle EDF = θA , angle ABC = angle DEF = θB and sides AC = DF, so the two triangles meet the
conditions for AAS. However, angle BCA = 180° - (θA + θB) = angle EFD, so the conditions for ASA are
also met, making the triangles congruent.
B
E
Case AAS = ASA
θB
θB
θA
θA
A
C
D
F
However, the ASS (or more politely, SSA) case with two pairs of corresponding sides equal and one
pair of equal angles, not included between the equal sides, is not a sufficient condition for
congruence. In this case, we see:
B
Case ASS (or SSA)
A
D
C
Comparing triangles ABD and ABC, we see that they share a common angle BAD, a common side AB
and that corresponding sides BD and BC are equal, the equal angle not included between the pairs of
equal sides. Thus, the two triangles, whilst clearly different, nevertheless meet the conditions of ASS.
Note that the triangle BDC is isosceles so that angles BDC and BCD are equal, making the interior
angle ADB equal to (180° – angle BDC) and thus (180° - BCD). This is important because with the two
sides and non-included angle we have sufficient information to find the value of the angle opposite
the other given side using the Sine Rule from trigonometry.
This might seem to give a single value for angle ACB but remember that:
Bill Bavington
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Equation 1
so giving the alternate value of angle ADB. It also explains why one special case of ASS gives
congruence, that of RHS, since obviously:
in which case points C and D would collapse together making BC (and BD) perpendicular to AC.
A similar situation exists when the area and two sides of a triangle are given and the angle between
the two sides in question is to be found. Here we use single letter trigonometric rather than threeletter geometric notation for angles. The general formula is:
and we are given the Area, a and b. Substituting in and solving for C will give two results, the acute
angle solution C and the corresponding obtuse angle solution (180° - C). Geometrically, this
corresponds to:
The ambiguous case given the area and two sides
D
A
b
B
b
a
C
E
and AC and DC are equal to . Angles ACB and DCE are equal, making angle BCD =
(180° - angle ACB). The vertical height of the two triangles are given by
and
and are thus equal given equation 1 above.
Triangles ABC and BCD have the same area since they share a common base and vertical height and
so meet the required condition for equal area.
BC equals
Bill Bavington
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