What if you were given a triangle with a line segment drawn through it from one side to the other? How could you use information about the triangle's side lengths to determine if that line segment is parallel to the third side? After completing this Concept, you'll be able to answer questions like this one.

### Watch This

CK-12 Foundation: Triangle Proportionality

First watch this video.

James Sousa: Triangle Proportionality Theorem

Now watch this video.

James Sousa: Using the Triangle Proportionality Theorem to Solve for Unknown Values

### Guidance

Think about a **midsegment** of a triangle. A midsegment is parallel to one side of a triangle and divides the other two sides into congruent halves. The midsegment divides those two sides **proportionally.** But what about another line that is parallel, but does not divide the other two sides into congruent halves? In fact, such a line will still divide the sides proportionally. This is called the **Triangle Proportionality Theorem.**

**Triangle Proportionality Theorem:** If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

If \begin{align*}\overline{DE} \| \overline{AC}\end{align*}, then @$\begin{align*}\frac{BD}{DA} = \frac{BE}{EC}\end{align*}@$. (@$\begin{align*}\frac{DA}{BD} = \frac{EC}{BE}\end{align*}@$ *is also a true proportion.*)

The converse of this theorem is also true.

**Triangle Proportionality Theorem Converse:** If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

If @$\begin{align*}\frac{BD}{DA} = \frac{BE}{EC}\end{align*}@$, then @$\begin{align*}\overline {DE} \| \overline{AC}\end{align*}@$.

#### Example A

A triangle with its midsegment is drawn below. What is the ratio that the midsegment divides the sides into?

The midsegment splits the sides evenly. The ratio would be 8:8 or 10:10, which both reduce to 1:1.

#### Example B

In the diagram below, @$\begin{align*}\overline {EB} \| \overline {CD}\end{align*}@$. Find @$\begin{align*}BC\end{align*}@$.

To solve, set up a proportion.

@$$\begin{align*}\frac{10}{15} = \frac{BC}{12} \longrightarrow \ 15(BC) &= 120\\ BC &= 8\end{align*}@$$

#### Example C

Is @$\begin{align*}\overline{DE} \| \overline{CB}\end{align*}@$?

If the ratios are equal, then the lines are parallel.

@$$\begin{align*}\frac{6}{18} = \frac{8}{24} = \frac{1}{3}\end{align*}@$$

Because the ratios are equal, @$\begin{align*}\overline {DE} \| \overline{CB}\end{align*}@$.

CK-12 Foundation: Triangle Proportionality

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### Guided Practice

Use the diagram to answers questions 1-5. @$\begin{align*}\overline{DB} \| \overline{FE}\end{align*}@$.

1. Name the similar triangles. Write the similarity statement.

2. @$\begin{align*}\frac{BE}{EC} = \frac{?}{FC}\end{align*}@$

3. @$\begin{align*}\frac{EC}{CB} = \frac{CF}{?}\end{align*}@$

4. @$\begin{align*}\frac{DB}{?} = \frac{BC}{EC}\end{align*}@$

5. @$\begin{align*}\frac{FC+?}{FC} = \frac{?}{FE}\end{align*}@$

**Answers:**

1. @$\begin{align*}\triangle DBC \sim \triangle FEC\end{align*}@$

2. DF

3. DC

4. FE

5. DF; DB

### Explore More

Use the diagram to answer questions 1-7. @$\begin{align*}\overline{AB} \| \overline {DE}\end{align*}@$.

- Find @$\begin{align*}BD\end{align*}@$.
- Find @$\begin{align*}DC\end{align*}@$.
- Find @$\begin{align*}DE\end{align*}@$.
- Find @$\begin{align*}AC\end{align*}@$.
- What is @$\begin{align*}BD:DC\end{align*}@$?
- What is @$\begin{align*}DC:BC\end{align*}@$?
- Why @$\begin{align*}BD:DC \neq DC:BC\end{align*}@$?

Use the given lengths to determine if @$\begin{align*}\overline{AB} \| \overline{DE}\end{align*}@$.