Similar triangles have
the same shape, but the size may be different.
Remember "@" means "is
congruent to" and "~" is "similar to".
Examples
Corresponding Triangles
Corresponding Congruent
Angles
Corresponding Proportional Sides
a/f = b/d = c/e = factor
DABC ~ DFDE
<A @ <F
<B @ <D
<C @ <E
a/f = 6/3 = 2
b/d = 8/4 = 2
c/e = 10/5 = 3
DABC @ DFDE
<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 or B. 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.
Congruent triangles are
a special type of similar triangles. Congruent triangles have
the same shape (similar triangles) and size.
Increase/decrease <A and <B
by clicking and dragging the vertices of the left triangle below.
Notice the corresponding angles (<D and
<E) remain congruent. Since the sum of three
angles must be 180o, the third pair of corresponding
angles must also be congruent when the first two pairs of corresponding
angles are congruent.
In the applet above:
<A @ <D, <B @ <E,
<C @ <F
Notice the size and shape of the new pairs
of triangles remains the same. The patterns you
may have observed and need to know for congruent triangles are
displayed below. Remember "@"
means "is congruent to".
Two triangles are congruent if:
all 6 pairs of corresponding angles and sides are congruent.
Triangles
Corresponding Congruent
Angles
Corresponding
Congruent Sides
factor = 1
a/f = b/d = c/e = 1
DABC
@ DDEF
<A @
<D
<B @ <E
<C @ <F
AB @
DE
BC @ EF
AC @ DF
The factor for congruent triangles
is 1. Remember this fact for future problem solving activities.
If all 3 pairs of corresponding sides in two triangles
are the same, then the triangles are congruent.
Move the corners A, B or C of the triangle
above. Watch how DDEF
changes. Since the 3 pairs of corresponding sides of the
two triangles are equal each time you manipulate the applet,
the SSS (Side, Side, Side) Congruence Relation
proves each pair of triangles are congruent.
Triangle
Congruence Relations
This is a prototype of JavaSketchpad,
a World-Wide-Web component of The
Geometer's Sketchpad. Copyright & copy ; 1990-1998
by Key Curriculum Press, Inc. All rights reserved. Portions
of this work were funded by the National Science Foundation
(awards DMI 9561674 & 9623018).
This page is best viewed by IE4 or Netscape Communicator (java-capable browsers).
SSS
Try to connect the points labeled B in the broken triangles. Can you make triangles
that aren't congruent?
If three sides of one triangle are
congruent to three sides of a
second triangle, the two triangles are congruent.
Try to connect the points labeled C in the broken triangles. Can you make triangles
that aren't congruent?
If two angles and the included
side of one
triangle are congruent to two angles and the included side of another triangle, the
triangles are congruent.
Try to connect the points labeled C in the broken triangles. Can you make triangles
that aren't congruent?
If two angles and a non-included
side of one triangle are congruent to two angles and the
corresponding nonincluded side of another triangle, the
two triangles are congruent.
If the hypotenuse and the leg of one right triangle are
congruent to the corresponding parts of the second right triangle, the two triangles are
congruent
SSA
How many triangles can be formed given two sides and the angle not between them?
Try to connect the points labeled B in the broken triangles. Can you create two
non-congruent triangles?
Two triangles with two sides and a non-included angle
equal may or may not be congruent.
How many triangles can be formed given three angles?
Drag points A and B in the triangle. Does the triangle maintain its shape and size?
(Angles A and B determine angle C)
If two angles on one triangle are
equal, respectively, to two angles on another triangle, then the triangles are similar,
but not necessarily congruent.
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.
two parallel lines (p & q)
and a transversal (t)
Before the next rules can apply, there must be two parallel
lines and a transversal.
symbols:
< angle @ congruent
Each time the transversal t
crosses a parallel line, two sets of opposite angles are formed.
Line p:
<1 @
<4 and <2 @ <3
Line q:
<5 @
<8 and <6 @ <7
Corresponding angles are on the same side of the transversal t..
Left side of transversal t:
<1 @
<5 and <3 @ <7
Right side of transversal t:
<4 @
<8 and <2 @ <6
Interior (inside) angles are the ones between the two parallel lines p and q.
<3 @ <6 and <4 @ <5
Exterior (outside) angles are the ones above and below the two parallel
lines p and q.
<1 @ <8 and <2 @ <7
If the angles in two triangles are
different, the triangles are neither similar nor congruent.
the same, the triangles are similar (AAA).
the same and the corresponding sides are the same size (SSS), the triangles are
congurent.
