Transcript 7-4: Parallel Lines and Proportional Parts

7-4: Parallel Lines and
Proportional Parts
Expectation:
G1.1.2: Solve multi-step problems and construct proofs
involving corresponding angles, alternate interior angles,
alternate exterior angles, and same-side (consecutive)
interior angles.
G1.2.2: Construct and justify arguments and solve multistep problems involving angle measure, side length,
perimeter, and area of all types of triangles.
G2.3.4: Use theorems about similar triangles to solve
problems with and without use of coordinates.
Given that ΔXYZ is similar to ΔABC, what is
the sum of the lengths of sides XY and XZ
in ΔXYZ below?
C
Z
3
A. 9
9
5
B
B. 12
y
4
Y
A
C. 15
z
D. 24
X
E. 27
Triangle Proportionality Theorem
(Side Splitting Theorem)
If a line is parallel to a side of a triangle and
intersects the other 2 sides in distinct points, then
it splits those 2 sides into segments of proportional
length.
Triangle Proportionality Theorem
B
D
A
E
C
AD
CE
If DE || AC,
=
BD
BE
4 common proportions from the
side splitting theorem
w
x
y
z
Determine the value of x below
if EB || CD.
A
5
E
3
D
x
B
4
C
In the diagram below, segments that
appear to be parallel are parallel. Solve
for x.
x
12
24
15
In the figure below, segment AB is parallel
to segment CD. What is the sum of AC
and BD?
A
A. 3⅓
C 8
B. 4⅓
3
2
C. 5⅓
D. 7⅔
B
E
D
E. 8⅔
5
Converse of Triangle
Proportionality Theorem
If a line intersects two sides of a
triangle in distinct points and
separates the sides into proportional
segments, then the line is parallel to
the third side of the triangle.
C
B
A
D
E
If AB = DE , then AE || BD.
BC
DC
Is AC || DE?
D
A
2
8
B
C
6
E
6
True or False? Justify
If BE = 3/5AB, then DC = 2/5BC.
B
E
D
C
A
Midpoint Connector Theorem
If a segment’s endpoints are the
midpoints of 2 sides of a triangle,
then it is parallel to and half the
length of the third side.
Midpoint Connector Theorem
B
E
A
D
C
E and D are the midpoints of their
respective sides so ED || AB and
AB = 2ED
Proportional Transversals
Corollary
If 3 or more parallel lines intersect 2
transversals, then they cut the
transversals proportionally.
Proportional Transversals
Corollary
s
t
y
a
b
l
z
m
n
a
y
If l || m and m || n, then
.
=
b
z
Find the values of all of the variables
below if w + x + y + z = 20
w
4
x
5
y
3
z
6
Congruent Transversal Segment
Theorem
If 3 or more parallel lines cut off congruent
segments on one transversal, then they
cut congruent segments on every
transversal.
Solve for x
l
x
m
12
u
n
t
Assignment
pages 367 – 369,
# 13 -29 (odds), 39 and 41 (odds)