Proportional Parts of Triangles

Let us learn about proportional parts of triangles,
Consider Figure 1 of Δ ABC with line l parallel to AC and intersecting the other two sides at D and E.

Figure 1 Deriving the Side-Splitter Theorem.

You can eventually prove that Δ ABC∼ Δ DBE using the AA Similarity Postulate. Because the ratios of corresponding sides of similar polygons are equal, you can show that



Now use Property 4, the Denominator Subtraction Property.




But AB–DB = AD, and BC–BE = CE ( Segment Addition Postulate). With this replacement, you get the following proportion.




This leads to the following theorem.
Theorem (Side-Splitter Theorem): If a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally.
Example 1: Use Figure 2 to find x.

Figure 2 Using the Side-Splitter Theorem.

Because DE ‖ AC in Δ ABC by Theorem 57, you get
Hope the above explanation was helpful.