Parallel Lines and Transversals . Angles and parallel lines.
Parallel Lines and Proportional Parts
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Parallel Lines and Proportional Parts
Chapter 7-4
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• midsegment
• Use proportional parts of triangles.• Divide a segment into parts.
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)
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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.
• The converse is true also.
B
A
E
DC
then DE // ifBEAB
CDACCB
DE //then , if CBBEAB
CDAC
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Example #1?DE // IsCB
DE //then , If CBBEAB
CDAC
B
A
E
DC24
26
9.75
9
75.9
26924
9(26) 4(9.75)2
234342
DE // Yes CB
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Find the Length of a Side
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Find the Length of a Side
Substitute the known measures.
Cross products
Multiply.
Divide each side by 8.
Simplify.
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A. 2.29
B. 4.125
C. 12
D. 15.75
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Determine Parallel Lines
In order to show that we must show that
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Determine Parallel Lines
Since the sides have
proportional length.
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1. A2. B3. C
A. yes
B. no
C. cannot be determined
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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
E
B
D
A
DE // AB
and
DE = AB21
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Midsegment of a Triangle
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Midsegment of a Triangle
Answer: D (0, 3), E (1, –1)
Use the Midpoint Formula to find the midpoints of
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Midsegment of a Triangle
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Midsegment of a Triangle
slope of
If the slopes of
slope of
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Midsegment of a Triangle
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Midsegment of a Triangle
First, use the Distance Formula to find BC and DE.
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Midsegment of a Triangle
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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)
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1. A2. B
A. yes
B. no
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1. A2. BA. yes
B. no
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Parallel Proportionality Theorem• If 3 // lines intersect two
transversals, then they divide the transversals proportionally.
then EF // CDAB// ifDFBD
CEAC
B
A
FD
C E
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Example #2
P 9
UTS
QR
15
11
Find ST
SP // TQ // UR
Corresponding Angle Thm.
11915 x
Parallel Proportionality Theorem
355
91651659
x
x
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Example #4J
K
M N
L7.5
9
13.5 x
y
37.5
Solve for x and y
What is JL? 37.5 – x
Solving for x
xx
5.37
5.139
)5.37(5.139 xx xx 5.1325.5069
25.5065.22 x5.22x
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Example #4J
K
M N
L7.5
9
13.5 x
y
37.5
Solve for x and ySolving for yJKL~JMN
AA~Theorem
y5.22
5.79
75.1689 y75.18y
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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
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Notice that the streets form a triangle that is cut by parallel lines. So you can use the Triangle Proportionality Theorem.
Answer: 32
Proportional Segments
Triangle Proportionality TheoremCross products
Multiply.
Divide each side by 13.
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A. 4
B. 5
C. 6
D. 7
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.
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Find x and y.
To find x:
Congruent Segments
GivenSubtract 2x from each side.Add 4 to each side.
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To find y:
Congruent Segments
The segments with lengths are congruent
since parallel lines that cut off congruent segments on
one transversal cut off congruent segments on every
transversal.
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Answer: x = 6; y = 3
Congruent Segments
Equal lengths
Multiply each side by 3 to eliminate the denominator.
Subtract 8y from each side.
Divide each side by 7.
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Find a.
A.
B. 1
C. 11
D. 7
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A. 0.5
B. 1.5
C. –6
D. 1
Find b.
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HomeworkHomeworkChapter 7-4Chapter 7-4
•Pg 410Pg 41013-21, 26 – 13-21, 26 –
27, 32 – 36, 6127, 32 – 36, 61