Three Theorems Involving Proportions Section 8.5

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

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)

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

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

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

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)

a G ZS Y I P X W b c d Given: a, b, c and d are parallel lines, Lengths given, WZ = 15 Find: WX, XY and YZ 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