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Three Theorems Involving Proportions

Explore three theorems involving proportions in triangles, including the Side Splitter Theorem, Angle Bisector Theorem, and Transversal Theorem. Use these theorems to solve problems and find length ratios in similar triangles.

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Three Theorems Involving Proportions

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  1. Three Theorems Involving Proportions Section 8.5

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

  3. 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)

  4. 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 =

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

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

  7. 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)

  8. a 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

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