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.

example

Find x.

sideEx
Apply the Side Splitter Theorem:
sideex1 (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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Theorem2
(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.
sideGiven
side1ab
While this theorem may look somewhat like the "mid-segment" theorem, the segment de2 in this theorem does not necessarily connect the "midpoints" of the sides.
Proof:
Statements
Reasons
sidepf1
1. Given
siddepf22
2. If 2 || lines are cut by a transversal, the corresponding angles are congruent.
sidepf3
3. (AA) If two ∠s of one Δ are congruent to the corresponding ∠s of another Δ, the Δs are similar.
side4
4. Corresponding sides of similar triangles are in proportion.
side5
5. Segment Addition Postulate (or whole quantity equals the sum of its parts)
side6
6. Substitution
side77
7. In a proportion, the product of the means = the product of the extremes.
side8
8. Distributive property
side9
9. Subtraction
side1010
10. In a proportion, the product of the means = the product of the extremes.

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Theorem2
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.
side11
side1ab

Proof:

Statements
Reasons
side12
1. Given
side13
2. In a proportion, the product of the means = the product of the extremes.
side15
3. Reflexive (Identity)
side16
4. Addition
side17
5. Distributive property
side6
6. In a proportion, the product of the means = the product of the extremes.
side18
7. Segment Addition Postulate (or whole quantity equals the sum of its parts)
side7
8. Substitution
side19
9. Reflexive (Identity)

 

side20
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.
side10
11. Corresponding angles of similar triangles are congruent.
site22
12. If 2 lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel.


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