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Geometry
Lesson Notes 6.4A
____________________
Objective: Use proportional parts of triangles to solve problems.
Triangle Proportionality Theorem: If a line is parallel to one side of a triangle, then it
separates the two other sides into segments of proportional lengths.
Note: This proportion is not the same as the scale factor for the s.
A
Practice:
Which triangles are similar? Why?
What is the ratio of their sides?
30
20
What is the ratio of the segments?
B
D
28
10
15
C
E
42
Example 1 (p 308): Find the length of a Side
F
In DFG, EH || FG , FE = 9, ED = 21,
and HG = 6. Find DH.
E
D
H
G
Note: This can, of course, be solved using
corresponding sides of similar triangles.
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Converse of the Triangle Proportionality Theorem: If a line separates two sides of a triangle
into corresponding segments of proportional lengths, then the line is parallel to the third side of
the triangle.
Example 2 (p 308): Determine if Lines are Parallel
In HKM, HM = 15, HN = 10,
and HJ is twice the length of JK .
Is MK || NJ ? Explain.
M
K
N
J
H
Midsegment of a triangle: a segment with its endpoints at the midpoints of two sides of a
triangle.
Theorem 6.6 Triangle Midsegment Theorem: A midsegment of a triangle is parallel to the
third side of the triangle and is half the length of the third side.
Practice:
x+6
3x − 8
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Example 3 (p 309): Midsegment of a Triangle on the Coordinate Plane
Triangle ABC has vertices A(–4, 1), B(8, –1), and C(–2, 9).
DE is a midsegment of ABC.
a. Find the coordinates of D and E.
b. Verify that AC is parallel to DE .
c. Verify that DE is ½AC.
 CW pp 311-312 #5-10
 HW
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A4a pp 312-313 #14-25
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Geometry
Objective:
Lesson Plan 6.4B
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Divide a segment into congruent parts.
Corollary 6.1: If three or more parallel lines intersect two transversals, then they cut off the
transversals proportionally.
AB XY

AC XZ
A
AB XY

BC YZ
BC YZ

AC XZ
X
B
Y
C
Z
Notice also, that the properties of proportions allow us tomake many more true proportions.
IMPORTANT: You can not use the parallel segments in these ratios!!!
Practice:
What values could x and y have?
x
6
10
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y
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Example 4 (p310): Proportional Segments
The distance from A Street to E Street along X Avenue is 3800 ft. The distance between
the same two streets is 4430 ft. along Y Avenue. The distance from C Street to D Street
is 411 ft. along X Avenue. What is the distance between the two streets along
Y Avenue?
A Street
B Street
C Street
D Street
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YAvenue
XAvenue
E Street
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E
Corollary 6.2: If three or more parallel lines cut off congruent segments on one transversal,
then they cut of congruent segments on every transversal.
Practice:
3x – 8
x+y
2x + 5
 CW p 312 #11-13, 33-34
 HW
A4b fms-Geometry Worksheet 6.4
A4c Lesson 6-4 Skills Practice / Practice
Prepare for Quiz 6.3-6.4
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