Theorem: A line parallel to one side of a triangle divides the other two proportionately.

In the figure below, segment DE is parallel to segment BC and segment EF is parallel to AB:

The figure shows triangle ABC with segments DE and DF. Point D is on side AB, point E is on side AC, and point F is on side BC. Segment AD is 16, segment AE is 24, segment EC is 36, and segment FC is 27.

Which statement can be proved true using the given theorem?

Segment BD = 32
Segment BD = 36
Segment BF = 15
Segment BF = 18

Given the problem and the theorem that states a line parallel to one side of a triangle divides the other two sides proportionately, let's analyze the situation:

### Triangle ABC:
- \( DE \parallel BC \) and \( DF \parallel AB \)
- Given lengths:
  - \( AD = 16 \)
  - \( AE = 24 \)
  - \( EC = 36 \)
  - \( FC = 27 \)

### Step 1: Proportionality due to parallel lines \( DE \parallel BC \)
By the given theorem, since \( DE \parallel BC \):
\[
\frac{AD}{DB} = \frac{AE}{EC}
\]
We know \( AD = 16 \), \( AE = 24 \), and \( EC = 36 \). Let \( BD = x \).

Thus, the proportion is:
\[
\frac{16}{x} = \frac{24}{36}
\]
Simplifying the right side:
\[
\frac{16}{x} = \frac{2}{3}
\]
Cross-multiplying gives:
\[
2x = 48 \implies x = 24
\]
However, this is \( BD \), but no option matches \( BD = 24 \).

### Step 2: Proportionality due to parallel lines \( DF \parallel AB \)
Since \( DF \parallel AB \), we apply the proportionality theorem to \( BF \) and \( FC \):
\[
\frac{BF}{FC} = \frac{BD}{DA}
\]
Given \( FC = 27 \) and \( BD = x \) (which was calculated as \( BD = 24 \)):
\[
\frac{BF}{27} = \frac{24}{16} = \frac{3}{2}
\]
Let \( BF = y \), so:
\[
\frac{y}{27} = \frac{3}{2}
\]
Cross-multiplying gives:
\[
2y = 81 \implies y = 40.5
\]
Again, \( BF = 40.5 \) does not match any of the options.

Since there seems to be a miscalculation, let's double-check the answer by plugging the options into the proportionality equations:

### Final Check:
- \( \textbf{Option BF = 18} \)
\[
\frac{18}{27} = \frac{2}{3}
\]
Matches proportion, making \( BF = 18 \) correct.

Thus, **Segment BF = 18** is the correct answer.

---

Would you like further details or have any questions? Here are some related questions:

1. How does the parallel line theorem apply in different types of triangles?
2. Can this theorem be used to prove other types of geometric theorems?
3. What are the implications of this theorem for similar triangles?
4. How does changing the lengths of other sides affect the proportions?
5. How would the solution change if the segments were not parallel?

**Tip:** Always double-check proportional calculations by plugging results back into the original equation!

Learn how to apply the triangle proportionality theorem to solve for segment lengths in a triangle with parallel sides. Example problem involving segments DE and DF parallel to sides BC and AB, respectively.

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