1. Three Theorems Involving Proportions
NOTES 8.5
Three Theorems Involving Proportions

2. Theorem 65: If a line is ║to one side of a Δ and intersects the other 2 sides, it divides those 2 sides proportionally. (Side splitter theorem)

3. Prove similar triangles first
Given: AT ║ BN
FA = 12, FT = 15
AT = 14, AB = 8
Find: length BN & NT A T B N
Prove similar triangles first =
BN = 23 1/3
TN = 10 =

4. Theorem 66: If three or more parallel lines are intersected by two transversals, the parallel lines divide the transversals proportionally.

5. Substitute BD for AG and DF for GF, substitution or transitive
Given: AB CD EF
Conclusion:     = A B D C G E F
Draw AF
In Δ EAF, =
In Δ ABF, =
Substitute BD for AG and DF for GF,
substitution or transitive =

6. Theorem 67: If a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the adjacent sides. (Angle Bisector Theorem)

7. BC • AD = AB • CD Means Extreme
Given: ∆ ABD
AC bisects <BAD
Prove: B D C =
Set up Proportions = =
BC • AD = AB • CD Means Extreme
Therefore: = =

8. According to Theorem 66, the ratio of WX:XY:YZ is 2:3:4. P W
Given: a, b, c and d are parallel lines, Lengths given, WZ = 15
Find: WX, XY and YZ 2 I X b 3 G Y c 4 d S Z
According to Theorem 66, the ratio of WX:XY:YZ is 2:3:4.
Therefore, let WX = 2x, XY = 3x and YZ = 4x
2x + 3x + 4x = 15
9x = 15
x = 5/3
WX = 10/3 = 3 1/3
XY = 5
YZ = 20/3 = 6 2/3