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Transcript
Math 310
Section 10.4
Similarity
Similar Triangles
Def
ΔABC is similar to ΔDEF, written
ΔABC
~ ΔDEF, iff <A is congruent to <D, <B is
congruent to <E, <C is congruent to <F and
AB/DE = AC/DF = BC/EF
Ex
The two following
triangles are similar:
ΔABC ~ ΔDEF.
E
70°
8
4
B
4
A
2
70°
80°
30°
3
C
D
80°
30°
6
F
AA Property
Thrm
If two angles of one triangle are congruent,
respectively, to two angles of a second triangle,
then the triangles are similar. Denoted: AA
Note: sometimes called the AAA property.
Ex
B
4
85°
A 45°
6
Are the two triangles similar? If
they are find the remaining sides.
C
E
8
85°
6
D
45°
F
Theorem 10-4
Thrm
If a line parallel to one side of a triangle intersects
the other sides, then it divides those sides into
proportional segments.
C
E
D
A
B
Ex
B
m BE
= 1.68
m ED
m AB = 3.93 cm
m BE = 3.59 cm
A
= 1.68
m AC
E
m AC = 2.33 cm
m ED = 2.13 cm
C
m AB
D
Ex
Suppose line DE is parallel
to line segment BA in
triangle ABC. If ratio of
BD to DC is 2/3 and CE
is length 3, what is the
length of AE?
C
D
E
B
2
A
Theorem 10-5
Thrm
If a line divides two sides of a triangle into
proportional segments, then the line is parallel to
the third side.
C
E
D
A
B
Ex
Find the measures of all
the interior angles of
triangle ABC.
C
6
D
4
B
35°, 105 °, 40 °
3
35° 105°
2
A
E
Theorem 10-6
Thrm
If parallel lines cut off congruent segments on
one transversal, then they cut off congruent
segments on any transversal.
3
3
2
2
Ex
5
7
?
5
Given the three lines are
parallel, what is the length of
the segment next to the
question mark?
7
Triangle Midsegment
Def
The midsegment of a triangle connects the
midpoints of two adjacent sides of the triangle.
Midsegment Theorem
Thrm
The midsegment is parallel to the third side of the
trianlge and half as long.
F
I
J
G
H
Ex
F
60°
J
I
H
50°
8
G
50°, 60°, 70°, 4
Given that JI is the
midsegment of triangle
FGH, find all the interior
angles of the triangle and
the length of the
midsegment.
Theorem 10-8
If a line bisects one side of a triangle and is
parallel to a second side, then it bisects the third
side and therefore is a midsegment.
F
J
7
G
7
25°
I
25°
H
Ex
H
I
110 °
F
30°
40°
3
G
J
3
Is the segment JI the
midsegment of
triangle FGH?
Indirect Measurement
One practical use of these theorems is the ability
to measure objects and distances that would be
impossible or impractical to do directly.
Ex
Book: pg 689.