1. Parallel Lines and Proportional Parts

2. Triangle Proportionality Theorem (or Side Splitter Theorem)
If a line is parallel to one side of a triangle and intersects the other two sides, it divides those two sides proportionally.
If BE || CD, then

3. Ex. 1: If AB||ED, BD = 8, DC = 4, and AE = 12.
Find EC
EC = 6

4. Ex. 2 If UY|| VX, UV = 3, UW = 18, XW = 16.
Find YX.
YX = 3.2

5. SU = 4. 5

6. Converse of Triangle Proportionality Theorem
If a line intersects the other two sides and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side. If
then BE || CD.

7. Ex. 4 Determine if GH || FE. Justify
In triangle DEF, DH = 18, and HE = 36, and DG = ½ GF.
To show GH || FE,
Show
Let GF = x, then DG = ½ x.
Continue on next slide

8. YES! Since the sides are proportional, then GH || FE. Substitute
Simplify by cancelling out the x’s from the numerator and denominator of the first fraction
Are the two fractions equivalent?
YES!
Since the sides are proportional,
then GH || FE.

9. Triangle Midsegment Theorem
A midsegment of a triangle is parallel to one side of the triangle, and its length is one-half the length of that side.
If D and E are midpoints of AB and AC,
Then DE || BC and
DE = ½ BC

10. 10 = AB

11. FE = 9

12. Ex. 7
In the figure, DE and DF are midsegments of ΔABC.
Find BC
DF = ½ BC
8 = ½ BC
BC = 16

13. In the figure, DE and DF are midsegments of ΔABC.
Ex. 8
In the figure, DE and DF are midsegments of ΔABC.
Find DE.
DE = ½ AC
DE = ½(15)
DE= 7.5

14. In the figure, DE and DF are midsegments of ΔABC.
Ex. 9
In the figure, DE and DF are midsegments of ΔABC.
Alternate interior angles