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D. N. A. Are the following triangles similar? If yes, state the appropriate triangle similarity theorem. 9. 2). 1). 15. 12. 8. 3 ) Find the value of x and the length of PQ. Parallel Lines and Proportional Parts. Chapter 7-4. Use proportional parts of triangles. - PowerPoint PPT Presentation
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D. N. A.Are the following triangles similar? If yes, state the
appropriate triangle similarity theorem.
1)A
B
CD
E
2) XY
Z
R
TS
1512
9
8
3) Find the value of x and the length of PQ.
x10
208P
Q
RN
M
Parallel Lines and Proportional Parts
Chapter 7-4
• 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)
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 // ifBE
AB
CD
ACCB
DE //then , if CBBE
AB
CD
AC
Example #1?DE // IsCB
DE //then , If CBBE
AB
CD
AC
B
A
E
DC24
26
9.75
9
75.9
26
9
24 9(26) 4(9.75)2
234342
DE // Yes CB
Find the Length of a Side
Substitute the known measures.
Cross products
Multiply.
Divide each side by 8.
Simplify.
Find the value of x and y.
1216
4y x
10
)1 )2
3612 8
y
x6
)3
15x
125
y
4
Determine Parallel Lines
Since the sides have
proportional length.
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
Midsegment of a Triangle
Answer: D (0, 3), E (1, –1)
Use the Midpoint Formula to find the midpoints of
Midsegment of a Triangle
First, use the Distance Formula to find BC and DE.
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)
Parallel Proportionality Theorem• If 3 // lines intersect two
transversals, then they divide the transversals proportionally.
then EF // CDAB// ifDF
BD
CE
AC
B
A
FD
C E
Example #2
P 9
UTS
QR
15
11
Find ST
SP // TQ // UR
Corresponding Angle Thm.
119
15 x
Parallel Proportionality Theorem
3
55
9
165
1659
x
x
Example #4
J
K
M N
L7.5
9
13.5x
y
37.5
Solve for x and y
What is JL? 37.5 – x
Solving for x
x
x
5.37
5.13
9
)5.37(5.139 xx xx 5.1325.5069
25.5065.22 x5.22x
Example #4
J
K
M N
L7.5
9
13.5x
y
37.5
Solve for x and ySolving for yJKL~JMN
AA~Theorem
y
5.22
5.7
9
75.1689 y75.18y
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.
Answer: 32
Proportional Segments
Triangle Proportionality Theorem
Cross products
Multiply.
Divide each side by 13.
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.
Find x and y.
To find x:
Congruent Segments
Given
Subtract 2x from each side.
Add 4 to each side.
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.
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.
HomeworkHomework
Chapter 7-4Chapter 7-4•Pg 410Pg 410
13-21, 26 – 13-21, 26 – 27, 32 – 36, 6127, 32 – 36, 61
