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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.
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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) |

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(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. |
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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 |
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1. Given |
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2. If 2 || lines are cut by a transversal, the corresponding angles are congruent. |
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3. (AA) If two ∠s of one Δ are congruent to the corresponding ∠s of another Δ, the Δs are similar. |
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4. Corresponding sides of similar triangles are in proportion. |
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5. Segment Addition Postulate (or whole quantity equals the sum of its parts) |
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6. Substitution |
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7. In a proportion, the product of the means = the product of the extremes. |
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8. Distributive property |
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9. Subtraction |
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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. |
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Proof:
Statements |
Reasons |
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1. Given |
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2. In a proportion, the product of the means = the product of the extremes. |
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3. Reflexive (Identity) |
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4. Addition |
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5. Distributive property |
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6. In a proportion, the product of the means = the product of the extremes. |
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7. Segment Addition Postulate (or whole quantity equals the sum of its parts) |
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8. Substitution |
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9. Reflexive (Identity)
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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. |
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11. Corresponding angles of similar triangles are congruent. |
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12. If 2 lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel. |

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