# Geometry: Parallel Segments and Segment Proportions

## Parallel Segments and Segment Proportions

Suppose you have a segment AB. You've talked about breaking up a segment into two equal pieces (using the midpoint). You can also break up a segment into thirds, quarters, or whatever fraction you want.

Suppose you have two segments, AB and RS, as shown in Figure 14.3. These segments can have the same length, or they can have different lengths. You can break each segment up into a variety of pieces. One specific way you can divide up your segments is proportionally. When two segments, AB and RS, are divided proportionally, it means that you have found two points, C on AB and T on RS, so that

• AC/RT = CB/TS. Figure 14.3AB and RS are divided proportionally, so that AC/RT = CB/TS

You are now ready to prove the following theorem:

• Theorem 14.2: If a line is parallel to one side of a triangle and intersects the other two sides, then it divides these sides proportionally.
• Example 1: Write a formal proof of Theorem 14.2.
• Solution: This theorem is illustrated in Figure 14.4. Figure 14.4?ABC has ?DE ? ? BC , with ?DE intersecting AB at D and AC at E.

• Given: In Figure 14.4, ?ABC has ?DE ? ? BC , with ?DE intersecting AB at D and AC at E.
• Proof: In order to show that D and E divide the segments AB and AC proportionally, you will need to show that ?ADE ~ ?ABC and then use CSSTAP. To show that ?ADE ~ ?ABC, you will use the AA Similarity Theorem. To determine the angle congruencies, you will use our postulate about corresponding angles and parallel lines.
StatementsReasons
1. ?ABC has ?DE ? ? BC , with ?DE intersecting AB at D and AC at E Given
2.?DE ? ? BC cut by a transversal AB Definition of transversal
3. ?ADE and ?ABC are corresponding anglesDefinition of corresponding angles
5. ?DAE ~ ?ABC Reflexive property of ~=
6. ?ADE ~ ?ABCAA Similarity Theorem
8. AB - AD/AD = AC - AE/AE Property 3 of proportionalities
10. AD/BD = AE/EC Property 2 of proportionalities

You can also use similar triangles to show that two lines are parallel. For example, suppose that ?ADE ~ ?ABC in Figure 14.5. You can prove that DE ? ? BC.

• Example 2: If ?ADE ~ ?ABC as shown in Figure 14.5, prove that DE ? ? BC. • Solution: Your game plan is quite simple. Because ?ADE ~ ?ABC, you know that ?ADE ~= ?ABC. Because ?ADE and ?ABC are congruent corresponding angles, you know that ?DE ? ? ?BC by Theorem 10.7.
StatementsReasons 