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Similar triangles
http://topdrawer.aamt.edu.au/Geometric-reasoning/Big-ideas/Similarity/Similartriangles
• Two triangles with the same shape are similar.
• Similarity is the result of an enlargement transformation and could also involve
rotations, translations and reflections.
• When naming pairs of similar triangles, the order of the letters naming the triangles
must be in matching order.
Tests for similar triangles
AAA
If two angles of one triangle are respectively equal to two angles of another triangle,
then the two triangles are similar.
P
A
Θ
•
B
C
Q
ΔABC ~ ΔPQR
Θ
•
R
( AAA )
When using this test, it is sufficient to prove that just two pairs of angles are equal. The
third pair of angles must then also be equal since the angle sum of any triangle is 180°.
SSS
If each of the sides of one triangle can be matched up with each of the sides of another
so that the ratios of matching sides are equal, then the two triangles are similar.
P
B
a
c
A
C
b
ΔABC ~ ΔQPR
ka
kc
Q
kb
R
(SSS)
AAMT — TOP DRAWER TEACHERS
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be used, reproduced, published, communicated and adapted free of charge for non-commercial
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page 1 of 2
SAS
If the ratio of the lengths of two sides of one triangle is equal to the ratio of the lengths
of two sides of another triangle, and the included angles are equal, then the two
triangles are similar.
B
P
ka
•
C
kb
A
ΔBAC ~ ΔPQR
a
•
Q
R
b
(SAS)
RHS
If the ratio of the hypotenuse and one side of a right-angled triangle is equal to the ratio
of the hypotenuse and one side of another right-angled triangle, then the two triangles
are similar.
R
B
ka
a
A
b
C
ΔACB ~ ΔPQR
P
(RHS)
AAMT — TOP DRAWER TEACHERS
page 2 of 2
kb
Q