Overview

A line parallel to one side of a triangle creates smaller similar triangles and proportional segments.

Key Ideas

• If $DE \parallel BC$ in $\triangle ABC$, then $\frac{AD}{DB} = \frac{AE}{EC}$.
• The smaller triangle is similar to the original.
• Parallel lines preserve angle measures, giving AA similarity quickly.

Core Skills

Use Similarity First

Once you identify parallel lines, write the similarity and then set up ratios. The proportionality statement becomes automatic.

Convert Between Whole and Parts

If you know $AD$ and $DB$, then $AB=AD+DB$. Be explicit about which side is the whole to avoid swapping ratios.

Extend to Multiple Parallels

If several parallel lines cut two sides, the consecutive segments are proportional across the two sides.

Worked Example

In $\triangle ABC$, $DE \parallel BC$, $AD=4$, $DB=2$, and $AE=6$. Find $EC$.

$AD/DB = AE/EC$ gives $4/2 = 6/EC$, so $EC=3$.

More Examples

Example 1: Find a Whole Side

If $AD=3$, $DB=5$, and $DE \parallel BC$, find $AB$.

$AB = 3+5 = 8$.

Example 2: Ratio from Similarity

If $DE \parallel BC$ and $AD/AB=2/5$, find $AE/AC$.

By similarity, $AE/AC=AD/AB=2/5$.

Strategy Checklist

• Confirm the line is parallel before using proportionality.
• Write the similarity statement to fix correspondence.
• Use whole-vs-part relationships carefully.

Common Pitfalls

• Using the proportionality theorem when the line is not parallel.
• Mixing segment ratios across different sides.
• Forgetting to compare corresponding sides (small to large consistently).

Practice Problems