Proportional Parts
Advanced GeometrySimilarityLesson 4
If a line is parallel to one side of a triangle and intersects the other two sides in two distinct points, then it separates
these sides into segments of proportional lengths.
Triangle Proportionality Theorem
If ,BD AE
CB CD.
BA DE
endpoints are the midpoints of two sides
Midsegment
Triangle Midsegment TheoremA midsegment of a triangle is parallel to one side of the triangle,
and its length is one-half the length of that side.
BD AE
1
2BD AE
and
Example: Find x, BD, and AE.
Proportional SegmentsIf three or more parallel lines intersect two transversals,
then they cut off the transversals proportionally.
FJ GK HL������������������������������������������������������������������ �����
FG JK
GH KL
Example: Find x.
If two triangles are similar, then their perimeters areproportional to the measures of the corresponding sides.
Proportional Perimeters
EXAMPLE:If ∆DEF ∆GFH, find the perimeter of ∆DEF.∼
If two triangles are similar, then the measures of the corresponding altitudes, angle bisectors, and medians
are proportional to the measures of the corresponding sides.
Special Segments of Similar Triangles
EG
JL
EXAMPLE: In the figure, ∆EFD ~ ∆JKI. is a median of ∆EDF and
is a median of ∆JIK. Find JL if EF = 36, EG = 18, and JK = 56.
EXAMPLE: The drawing below illustrates two poles supported by wires. ∆ABC ~ ∆GED. CFAF FG GC DC.
EC. and
Find the height of pole
An angle bisector in a triangle separates the opposite side intosegments that have the same ratio as the other two sides.
Angle Bisectors
AD
AB
segments with
endpoint A
segments with
endpoint C
CD
CB
EXAMPLE:Find x if AB = 10, AD = 6, DC = x, and BC = x + 6.