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Congruent means being exactly the same. When two line segments have the
same length, they are congruent. When two figures have the same shape and size,
they are congruent.
These two triangles are congruent. They are exactly the same in every way.
They are the same size and the same shape. Their side lengths are the same and that
their angle measures are the same. Sometimes, two figures will be similar. Similar
means that the figures have the same shape, but not the same size. Similar figures
are not congruent.
Have you ever seen siblings who look so alike that one seems to be a miniature
copy of the other? Similar polygons follow the same concept of being the same shape
but different in size. We can prove similarity in polygons, and there are several ways
to prove that triangles are similar. Similar polygons also have many properties and
relationships that can be used to solve problems.
Let us now learn the Basic Proportionality Theorem, its converse, and
segments cut by transversals and by angle bisector.
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Illustration: In ∆XYZ, P and Q are points on XY and XZ respectively,
such that PQ ∥ YZ.
By applying BPT, we can formulate the
following proportions:
XP
=
XQ
OR
XP
=
PY
PY QZ XQ QZ
Example 1: In the given figure, if XP = 6 cm, YP = 2 cm, XQ = 7.5 cm, find QZ.
