Transcript Geo Ch 7-4 – Parallel Lines and Proportional Parts

Parallel Lines and
Proportional Parts
Chapter 7-4
• Use proportional parts of triangles.
• Divide a segment into parts.
• midsegment
Standard 12.0 Students find and use measures of
sides and of interior and exterior angles of triangles
and polygons to classify figures and solve problems.
(Key)
Triangle Proportionality Theorem
• If a line parallel to one side of
a triangle intersects the other
two sides, then it divides the A
two sides proportionally.
• The converse is true also.
AC AB
if CB // DE then

CD BE
AC AB
if

, then CB // DE
CD BE
C
D
B
E
Example #1
Is CB // DE?
A
AC AB
If

, then CB // DE
CD BE
24 26
24(9.75)  9(26)

9 9.75
24
C 9
D
26
234  234
Yes CB // DE
B
9.75
E
Find the Length of a Side
Find the Length of a Side
Substitute the known measures.
Cross products
Multiply.
Divide each side by 8.
Simplify.
A. 2.29
B. 4.125
C. 12
D. 15.75
Determine Parallel Lines
In order to show that
we must show that
Determine Parallel Lines
Since
proportional length.
the sides have
A. yes
B. no
C. cannot be
determined
1.
2.
3.
A
B
C
Midsegment Theorem
• The midsegment connecting the midpoints
of two sides of the triangle is parallel to the
third side and is half as long.
C
DE // AB
and
D
A
1
DE = 2 AB
E
B
Midsegment of a Triangle
Midsegment of a Triangle
Use the Midpoint Formula to find the midpoints of
Answer: D (0, 3), E (1, –1)
Midsegment of a Triangle
Midsegment of a Triangle
If the slopes of
slope of
slope of
Midsegment of a Triangle
Midsegment of a Triangle
First, use the Distance Formula to find BC and DE.
Midsegment of a Triangle
A. W (0, 1), Z (1, –3)
B. W (0, 2), Z (2, –3)
C. W (0, 3), Z (2, –3)
D. W (0, 2), Z (1, –3)
A. yes
B. no
1.
2.
A
B
A. yes
B. no
1.
2.
A
B
Parallel Proportionality Theorem
• If 3 // lines intersect two
transversals, then they
divide the transversals
proportionally.
A
B
AC BD
if AB// CD // EF then

CE DF
C
D
E
F
Find ST
Example #2
SP // TQ // UR
U
T
S
11
Corresponding Angle Thm.
15 x Parallel

Proportionality
9 11 Theorem
9 x  165
165 55
x

9
3
P
15
9
Q
R
Example #4
Solving for x
What is JL?
9
37.5  x

13.5
x
9 x  13.5(37.5  x)
9x  506.25 13.5x
22.5x  506.25
x  22.5
Solve for x and y
J
9
37.5
K
7.5
L
x
13.5
M
y
N
Example #4
Solving for y
JKL~JMN
AA~Theorem
9
22.5

7.5
y
9 y  168.75
y  18.75
Solve for x and y
J
9
37.5
K
7.5
L
x
13.5
M
y
N
Proportional Segments
MAPS In the figure, Larch, Maple, and Nuthatch
Streets are all parallel. The figure shows the distances
in city blocks that the streets are apart. Find x.
Proportional Segments
Notice that the streets form a triangle that is cut by
parallel lines. So you can use the Triangle Proportionality
Theorem.
Triangle Proportionality
Theorem
Cross products
Multiply.
Divide each side by 13.
Answer: 32
In the figure, Davis, Broad, and Main Streets are all
parallel. The figure shows the distances in city
blocks that the streets are apart. Find x.
A. 4
B. 5
C. 6
D. 7
Congruent Segments
Find x and y.
To find x:
Given
Subtract 2x from each side.
Add 4 to each side.
Congruent Segments
To find y:
The segments with lengths
are congruent
since parallel lines that cut off congruent segments on
one transversal cut off congruent segments on every
transversal.
Congruent Segments
Equal lengths
Multiply each side by 3 to
eliminate the denominator.
Subtract 8y from each side.
Divide each side by 7.
Answer: x = 6; y = 3
Find a.
A.
B. 1
C. 11
D. 7
Find b.
A. 0.5
B. 1.5
C. –6
D. 1
Homework
Chapter 7-4
• Pg 410
13-21, 26 – 27,
32 – 36, 61