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Bell Work:
Solve the proportion
9 = 18
3
d
Answer:
d=6
Two figures are similar if they have
the same shape even though they
may vary in size. In the illustration,
triangles A, B and C are similar.
Triangle D is not similar because its
shape is different.
A
B
C
D
If figures are the same shape and
size, they are not only similar, they
are also congruent. All three
triangles below are similar, but only
triangles A and B are congruent.
A
B
C
When inspecting polygons to
determine whether they are similar
or congruent, we compare their
corresponding parts.
Corresponding
Angles
<A and <D
<B and <E
<C and <F
Corresponding
Sides
__ and DE
___
AB
___ and EF
___
BC
___ and FD
___
CA
A D
C
B E
F
We use tick marks on corresponding
side lengths that have equal length
and arcs on corresponding angles
that have the same measure.
The symbol ~ represents similarity.
Since triangle ABC is similar to GHI,
we can write
triangle ABC ~ triangle GHI
G
A
C
B
H
I
The symbol ≅ represents
congruence. Since triangle ABC and
triangle DEF are congruent, we can
write
triangle ABC ≅ triangle DEF
D
A
B
C
E
F
Notice that when we write a
statement about the similarity or
congruence of two polygons, we
name the letters of the
corresponding vertices in the same
order.
Similar Polygons: have
corresponding angles which are the
same measure and corresponding
sides which are proportional in
length.
Congruent Polygons: have
corresponding angles which are the
same measure and corresponding
sides which are the same length.
The following two quadrilaterals are
similar.
12
100°
8
80°
8
8
80°
100°
12
100°
80°
6
6
80°
100°
8
The corresponding angles are
congruent and the corresponding sides
are proportional
The relationship between the sides
and angles of a triangle is such that
the lengths of the sides determine
the size and position of the angles.
For example, with three straws of
different lengths, we can form one and
only one shape of triangle. Three other
straws of twice the length would form a
similar triangle with angles of the same
measure. Knowing that two triangle
have proportional corresponding side
lengths is enough to determine that
they are similar.
Side – Side – Side Triangle Similarity:
if two triangles have
proportional corresponding side
lengths, then the triangles are
similar.
A
D
6
4
C
8
12
8
B
F
16
E
Triangle ABC ~ Triangle DEF
Side – Side – Side
All triangles with the same set of
angle measures are similar. Thus,
knowing that two triangles have
congruent corresponding angles is
sufficient information to conclude
that the triangles are similar.
Angle – Angle – Angle Similarity:
if the angles of one triangle are
congruent to the angles of
another triangle, then the
triangles are similar and their
corresponding side lengths are
proportional.
B
E
100°
100°
A
20°
60°
20°
C
D
60°
F
Triangle ABC ~ Triangle DEF
Angle – Angle – Angle
Example:
The triangles are similar. Find x.
5
15
x
3
4
12
Answer:
x=9
Left
Triangle
3
4
5
Right
Triangle
x
12
15
Example:
An architect makes a scale drawing
of a building. If one inch on the
drawing represents 4 feet, then what
is the scale factor form the drawing
to the actual building?
Answer:
If 1 inch represents 4 feet, then 1
inch represents 48 inches.
Therefore, the scale is 1:48 and the
scale factor is 48.
Example:
An architect makes a scale drawing
of a building. What is the length and
width of a room that is 5 inches by 4
inches on the drawing?
Answer:
Each inch represents 4 feet, so 5
inches represents 5 x 4 = 20 feet, and
4 inches represents 4 x 4 = 16 feet.
The room is 20 feet long and
16 feet wide.
HW: Lesson 35 #1-30
Due Tomorrow