 "Side Splitter" Theorem MathBitsNotebook.com Terms of Use   Contact Person: Donna Roberts The "Side Splitter" Theorem says that if a line intersects two sides of a triangle and is parallel to the third side of the triangle, it divides those two sides proportionally. Find x. Apply the Side Splitter Theorem: (form a proportion using the side lengths) Solve the proportion for x: 4x = (2)(7) 4x = 14 x = 3.5 (Answer)  (Side Splitter Theorem): If a line is parallel to a side of a triangle and intersects the other two sides, then this line divides those two sides proportionally.  While this theorem may look somewhat like the "mid-segment" theorem, the segment in this theorem does not necessarily connect the "midpoints" of the sides.
Proof:
 Statements Reasons 1. Given 2. If 2 || lines are cut by a transversal, the corresponding angles are congruent. 3. (AA) If two ∠s of one Δ are congruent to the corresponding ∠s of another Δ, the Δs are similar. 4. Corresponding sides of similar triangles are in proportion. 5. Segment Addition Postulate (or whole quantity equals the sum of its parts) 6. Substitution 7. In a proportion, the product of the means = the product of the extremes. 8. Distributive property 9. Subtraction 10. In a proportion, the product of the means = the product of the extremes.  Converse (Side Splitter Theorem): If a line intersects two sides of a triangle and divides the sides proportionally, the line is parallel to the third side of the triangle.  Proof:

 Statements Reasons 1. Given 2. In a proportion, the product of the means = the product of the extremes. 3. Reflexive (Identity) 4. Addition 5. Distributive property 6. In a proportion, the product of the means = the product of the extremes. 7. Segment Addition Postulate (or whole quantity equals the sum of its parts) 8. Substitution 9. Reflexive (Identity) 10. (SAS for Similarity). In two triangles, if two sets of corresponding sides are proportional and the included angle is congruent, the triangles are similar. 11. Corresponding angles of similar triangles are congruent. 12. If 2 lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel. 