Warm Ups
1. Find the missing perimeter.
P = ? mm
3 6
P = 25 mm
2. Solve for x.
12
8
x
Section 6.6
Use Proportionality Theorems
Isn’t logic a
beautiful thing!
Activity
B
In the diagram, DE | | AC.
D
1. Name a pair of similar
A
triangles and explain
why they are similar.
E
C
Activity
B
In the diagram, DE || AC.
D
1. Name a pair of similar
triangles and explain
E
C
A
B
why they are similar.
B
D
E
A
C
Activity
B
3. Write a proportion & solve for x.
10
848
4.
What is the ratio BD: DA?
Reduce your answer.
5. What is the ratio BE: EC ?
Reduce your answer.
6. What do you notice?
60
D
E
636
x
C
A
8
86

10
10  x
Proportionality Theorems!
Theorem 6.4
Triangle
Proportionality
Theorem
If a line parallel to one side
of a triangle intersects
the other two sides, then
it divides the two sides
proportionally.
Example 1
Find the length of YZ.
Proportionality Theorems!
Theorem 6.5
Converse of the
Triangle
Proportionality
Theorem
If a line divides two sides
of a triangle
proportionally, then it is
parallel to the third side.
Example 2
Determine whether PS || QR.
Example 3
Find the value of
x so that
BC || ED .
Discussion
Recall that the distance
between two parallel
lines is always equal.
This distance, however,
must be measured
along a perpendicular
segment.
CD  EF
Discussion
But what if the distance is
not perpendicular?
D
A
Are these lengths still
equal? Or does some
E
B
other relationship
exist?
F
C
Proportionality Theorems!
Theorem 6.6
If three parallel lines
intersect two
transversals, then they
divide the transversals
proportionally.
Example 4
Find the length of AB.
Discussion
Notice that the angle
bisector also divides
Recall
angle
the that
thirdan
side
of the
bisector
a ray
triangle is
into
twothat
parts.
divides
an angle
Are those
parts into
two
congruentOr
parts.
congruent?
is there
some other
relationship between A
them?
B
D
C
Proportionality Theorems!
Theorem 6.7
Angle Bisector
Proportionality Theorem
If a ray bisects an angle of a
triangle, then it divides the
opposite side into segments
whose lengths are
proportional to the other two
sides.
Example 5
Find the value of x.
Example 6
Find the value of x.
Assignment