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Three Theorems Involving Proportions
NOTES 8.5 Three Theorems Involving Proportions
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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)
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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 =
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Theorem 66: If three or more parallel lines are intersected by two transversals, the parallel lines divide the transversals proportionally.
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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 =
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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)
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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: = =
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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
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