4. 4
INTRODUCTION
Natural resources occur naturally
within environments that exist relatively undisturbed by
humanity, in a natural form. A natural resource is often
characterized by amounts of biodiversity and geodiversity
existent in various ecosystems. In geometry, two figures or
objects are congruent if they have the same shape and size,
or if one has the same shape and size as the mirror image of
the other. Two geometrical objects are called similar if they
both have the same shape, or one has the same shape as
the mirror image of the other.
5. 5
CONTENT
In elementary geometry the
word congruent is often used as follows. The word equal is
often used in place of congruent for these objects.
Two line segments are congruent if they have the same
length.
Two angles are congruent if they have the same
measure.
Two circles are congruent if they have the same
diameter
.
6. 6
If two objects are similar, each is congruent to the
result of a uniform scaling of the other.
In this figuer they are congruent
Orange
Determining congruence of
polygons
For two polygons to be congruent, they
must have an equal number of sides (and hence an equal
number—the same number—of vertices). Two polygons
with n sides are congruent if they each have numerically
identical sequences (even if clockwise for one polygon
and counterclockwise for the other) side-angle-side-angle-...
for n sides and n angles.
7. 7
Congruence of polygons can be established graphically as
follows:
First, match and label the corresponding vertices of the
two figures.
Second, draw a vector from one of the vertices of the
one of the figures to the corresponding vertex of the
other figure. Translate the first figure by this vector so
that these two vertices match.
Third, rotate the translated figure about the matched
vertex until one pair of corresponding sides matches.
Fourth, reflect the rotated figure about this matched
side until the figures match.
If at anytime the step cannot be completed, the polygons
are not congruent.
Similarity
Two geometrical objects are
called similar if they both have the same shape, or one has
the same shape as the mirror image of the other.
For example, all circles are similar to each other, all squares
are similar to each other, and all equilateral triangles are
similar to each other. On the other hand, ellipses are not all
similar to each other, rectangles are not all similar to each
other, and isosceles triangles are not all similar to each
other.
8. 8
Two
geometrical
objects are
called
similar if
they both
have the
same shape,
or one has
the same
shape as the
Two
geometr
ical
objects
are
called
similar if
they
both
have the
same
shape,
or one
has the
same
shape as
These are similar circle.
Orange
mirror
image of the
other.
the
mirror
image of
the
other.
If two angles of a triangle have measures equal to the
measures of two angles of another triangle, then the
triangles are similar. Corresponding sides of similar
polygons are in proportion, and corresponding angles of
similar polygons have the same measure.
Figures shown in the same color are similar
9. 9
Conclusion
Two triangles are congruent if
their corresponding sides are equal in length and their
corresponding angles are equal in size. Either object can be
rescaled, repositioned, and reflected, so as to coincide
precisely with the other object. If two objects are similar,
each is congruent to the result of a uniform scaling of the
other. Two geometrical objects are called similar if they
both have the same Two geometrical objects are called
similar if they both have the same shape, or one has the
same shape as the mirror image of the other, or one has the
same shape as the mirror image of the other.
