# 7.5: Proportionality Relationships

## Learning Objectives

- Identify proportional segments within triangles.
- Extend triangle proportionality to parallel lines.

## Review Queue

- Write a similarity statement for the two triangles in the diagram. Why are they similar?
- If find .
- If , find .
- Find and .

**Know What?** To the right is a street map of part of Washington DC. Street, Street, and Street are parallel and 7 Street is perpendicular to all three. All the measurements are given on the map. What are and ?

## Triangle Proportionality

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.*

**Example 1:** A triangle with its midsegment is drawn below. What is the ratio that the midsegment divides the sides into?

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

*The midsegment divides the two sides of the triangle proportionally, but what about other segments?*

**Investigation 7-4: Triangle Proportionality**

Tools Needed: pencil, paper, ruler

1. Draw . Label the vertices.

2. Draw so that is on ̅and is on . and can be ** anywhere** on these sides.

3. Is ? Why or why not? Measure , and . Then set up the ratios and . Are they equal?

4. Draw a second triangle, . Label the vertices.

5. Draw so that is on and is on AND .

6. Is ? Why or why not? Measure and . Then set up the ratios and . Are they equal?

From this investigation, we see that if , then divides the sides proportionally.

**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 , then . ( *is also a true proportion.*)

For the converse:

If , then .

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

*Proof of the Triangle Proportionality Theorem*

Given: with

Prove:

Statement |
Reason |
---|---|

1. | Given |

2. | Corresponding Angles Postulate |

3. | AA Similarity Postulate |

4. | Segment Addition Postulate |

5. | Corresponding sides in similar triangles are proportional |

6. | Substitution PoE |

7. | Separate the fractions |

8. | Substitution PoE (something over itself always equals 1) |

9. | Subtraction PoE |

We will not prove the converse; it is basically this proof but in the reverse order.

**Example 2:** In the diagram below, . Find .

**Solution:** Set up a proportion.

**Example 3:** Is ?

**Solution:** If the ratios are equal, then the lines are parallel.

Because the ratios are equal, .

## Parallel Lines and Transversals

We can extend the Triangle Proportionality Theorem to multiple parallel lines.

**Theorem 7-7:** If three parallel lines are cut by two transversals, then they divide the transversals proportionally.

If , then or .

**Example 4:** Find .

**Solution:** The three lines are marked parallel, set up a proportion.

**Example 5:** Find .

**Solution:** Set up a proportion.

**Example 6:** ** Algebra Connection** Find the value of that makes the lines parallel.

**Solution:** Set up a proportion and solve for .

Theorem 7-7 can be expanded to ** any** number of parallel lines with

**number of transversals. When this happens all corresponding segments of the transversals are proportional.**

*any*
**Example 7:** Find and .

**Solution:** Line up the segments that are opposite each other.

## Proportions with Angle Bisectors

The last proportional relationship we will explore is how an angle bisector intersects the opposite side of a triangle.

**Theorem 7-8:** If a ray bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the lengths of the other two sides.

If , then .

**Example 8:** Find .

**Solution:** The ray is the angle bisector and it splits the opposite side in the same ratio as the sides. The proportion is:

**Example 9:** ** Algebra Connection** Find the value of that would make the proportion true.

**Solution:** You can set up this proportion like the previous example.

**Know What?** Revisited To find and , you need to set up a proportion using parallel the parallel lines.

From this, and .

## Review Questions

- Questions 1-12 are similar to Examples 1 and 2 and review.
- Questions 13-18 are similar to Example 3.
- Questions 19-24 are similar to Examples 8 and 9.
- Questions 25-30 are similar to Examples 4-7.

Use the diagram to answers questions 1-5. .

- Name the similar triangles. Write the similarity statement.

Use the diagram to answer questions 6-12. .

- Find .
- Find .
- Find .
- Find .
- What is ?
- What is ?
- Why ?

Use the given lengths to determine if .

** Algebra Connection** Find the value of the missing variable(s).

Find the value of each variable in the pictures below.

## Review Queue Answers

- by AA Similarity Postulate