In this section we will be studying
congruent (≅) and similar (~)
triangles. It will be easier to remember these concepts
if you start by considering size and shape.
Similar triangles have
the same shape, but the size may be different.
Remember "≅" means "is
congruent to" and "~" is "similar
Corresponding Proportional Sides
a/f = b/d = c/e = factor
ΔABC ~ ΔFDE
<A = <F
<B = <D
<C = <E
a/f = 6/3 = 2
b/d = 8/4 = 2
c/e = 10/5 = 2
ΔABC ≅ ΔFDE
<A = <F
<B = <D
<C = <E
a/f = 3/3 = 1
b/d = 4/4 = 1
c/e = 5/5 = 1
Two triangles are similar if:
two pairs of corresponding angles are congruent (therefore
the third pair of corresponding angles are also congruent).
the three pairs of corresponding sides are proportional.
Notice the corresponding angles for the two triangles
in the applet are the same. The corresponding sides lengths
are the same only when the scale factor slider is set
at 1.0. Study the side lengths closely and you will find
that the corresponding sides are proportional.
If you know triangles are similar,
you can use the proportion of corresponding sides to
help determine an unknown dimension.
Study the object below. You can change
the triangles by dragging on the slider or dragging vertex
A, B or C. The proportions of the corresponding pairs
of sides changes as the scale slider position changes. Each
time you stop dragging, look at the proportions. If
one of the sides was unknown you could use two pairs of
corresponding sides to calculate the missing dimension.
When you learn the mathematics of similar
triangles, you can you use the knowledge to determine unknown
dimensions or angles without measuring. You have already
explored the angle relationships of similar triangles.
If you know an angle in one triangle, the corresponding
angle in a similar triangle will be the same size.
In the past you studied, corresponding angles
when a line (transfersal) intersects parallel lines.
Review the chart to refresh your memory. Angles are
usually the key to determine if triangles are similar.
parallel lines (p & q)
and a transversal (t)
Before the next rules
can apply, there must be two parallel lines and
Each time the transversal
crosses a parallel line, two sets of opposite angles
and <2 = <3
and <6 = <7
are on the same side of the transversal t..
side of transversal t:
and <3 = <7
Right side of transversal t:
and <2 = <6
Interior (inside) angles are the ones
between the two parallel lines p and q.
<6 and <4 = <5
Exterior (outside) angles are the
ones above and below the two parallel lines p