7-4 parallel lines and proportional parts 7-4 parallel lines and proportional parts geometry
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7-4 Parallel Lines and 7-4 Parallel Lines and Proportional PartsProportional Parts
7-4 Parallel Lines and 7-4 Parallel Lines and Proportional PartsProportional Parts
GeometryGeometry
• Use proportional parts with triangles.
• Use proportional parts with parallel lines.
• Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Theorems 7.5 Triangle Proportionality
Theorem
Q
S
R
T
U
If a line parallel to one side of a triangle intersects the other two sides, then it divides the two side proportionally.
If TU ║ QS, thenRT
TQ
RUUS
=
Ex. 1 In ΔPQR, ST//RQ. If PT = 7.5, TQ = 3, and SR = 2.5, find PS.
P
S T
R Q
7.5
32.5
x
Ex. 2 AB is parallel to MN, find x.
A B
M N
10
5
8
x
Ex. 3 NP is parallel to RS, find x.
10 12
15 x
M
N P
R S
Ex. 4: Finding the length of a segment
• In the diagram AB ║ ED, BD = 8, DC = 4, and AE = 12. What is the length of EC?
128
4
C
B A
D E
Theorems 7.6 Converse of the Triangle
Proportionality Theorem
Q
S
R
T
U
If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
RT
TQ
RUUS
=If , then TU ║ QS.
Ex. 5: Determining Parallels
• Given the diagram, determine whether MN ║ GH.
21
1648
56
L
G
H
M
N
LMMG
56
21=
8
3=
LN
NH
48
16=
3
1=
8
3
3
1≠
MN is not parallel to GH.
Theorem 7.7 A midsegment of a triangle is parallel to one side of the triangle, and its length is one half the length of that side.
• AB = ½ PS
A B
P S
Proportional Parts of Parallel Lines
• If three parallel lines intersect two transversals, then they divide the transversals proportionally.
• If r ║ s and s║ t and l and m intersect, r, s, and t, then
UWWY
VX
XZ=
l
m
s
Z
YW
XV
U
rt
Ex. 3: Using Proportionality
Theorems
• In the diagram 1 2 3, and PQ = 9, QR = 15, and ST = 11. What is the length of TU?
11
15
9
3
2
1
S
T
UR
Q
P
PQ
QR
ST
TU=
9
15
11
TU=
9 ● TU = 15 ● 11 Cross Product property
15(11)
9
55
3=TU =
Parallel lines divide transversals proportionally.
Substitute
Divide each side by 9 and simplify.
So, the length of TU is 55/3 or 18 1/3.
• Class work page 495 • Problems 1-9• Homework on page 496• Problems 10-17