Corresponding parts in a figure
mean that the parts (angles or sides) are in the same relative position in
each of the figures. For example, in the figure below there are two triangles:
a larger triangle, ABC, and a smaller triangle, ADE. Angles ADE and ABC are
corresponding because they both lie in the same relative position in the two
triangles. Angles ABC and AED are not corresponding since they do not lie
in the same relative position in the two triangles. For the same reason, sides
AE and AC are corresponding while sides BC and AD are not.
Similar figures have the same
shape, but not always the same size. Similar figures and congruent figures
both have the same shape. The difference between similar figures and congruent
figures is that congruent figures also have the same size. So, corresponding
sides and corresponding angles of congruent figures have the same measures.
The measures of corresponding angles in similar figures are the same, but
the measures of corresponding sides in similar figures are multiples of each
other (i.e. they are proportional). Therefore, in order to prove that any
two figures are similar, one must show that these two conditions hold for
all corresponding angles and all corresponding sides between the two figures.
For example, the two parallelograms below are similar figures.
Notice that they have the same shape, but obviously not the same size. All
four of their corresponding angles are congruent, and all four of their corresponding
sides are proportional (here, in a ratio of 1 to 2, or 0.500).
As the simplest type of polygon, triangles are unique in
that there are certain properties that can be used to verify that two triangles
are similar. Instead of showing that all corresponding angles are congruent
and all corresponding sides are in proportion, one can simply use one
of the following three properties. You will notice that these properties seem
a little too familiar (see also Congruent
According to the AA similarity property, to show two triangles
are similar, it is sufficient to verify that two angles of one triangle are
congruent to the corresponding angles of another triangle. In other words,
this tells you that the shape of a triangle is completely determined by the
measures of its angles.
According to the SAS similarity property, to show two triangles
are similar, it is sufficient to verify that the measures of two sides of
a triangle are proportional to the measures of two corresponding sides of
another triangle, and the included angles are congruent.
According to the SSS similarity property, to show two triangles
are similar, it is sufficient to verify that the measures of the three sides
of one triangle are proprotional to the corresponding measures of the sides
of another triangle.
Check for Understanding
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External Links to Extend Basic Understanding
Animated Math Glossary: Read another definition of similarity.
Triangles at Math.com: See a lesson about similar triangles and then
test yourself with the included questions.
Related Investigations on InterMath to
Similar Triangles: Determine proportions
comparing the side lengths of similar triangles.
Side Splitter: Find a relationship between sides in two
Altitude to the Hypotenuse: Investigate the relationship
between sides of right triangles when an altitude is drawn to the hypotenuse
of a right triangle.
Fractal Iterations: Explore the idea of fractals.
Double the fun: Double the dimensions of quadrilaterals.
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