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 14.3¯AB 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 14.4ΔABC has ↔DE ‌ ‌ ¯BC , with ↔DE intersecting ¯AB at D and ¯AC at E.

Given: In Figure 14.4, ΔABC has ↔DE ‌ ‌ ¯BC , with ↔DE intersecting ¯AB at D and ¯AC at E.
Prove: ^AD/_DB = ^AE/_EC.
Proof: In order to show that D and E divide the segments ¯AB and ¯AC proportionally, you will need to show that ΔADE ~ ΔABC and then use CSSTAP. To show that ΔADE ~ ΔABC, you will use the AA Similarity Theorem. To determine the angle congruencies, you will use our postulate about corresponding angles and parallel lines.

	Statements	Reasons
1.	ΔABC 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 angles	Definition of corresponding angles
4.	∠ADE ~= ∠ABC	Postulate 10.1
5.	∠DAE ~ ∠ABC	Reflexive property of ~=
6.	ΔADE ~ ΔABC	AA Similarity Theorem
7.	^AB/^AD = ^AC/_AE	CSSTAP
8.	^{AB - AD}/_AD = ^{AC - AE}/_AE	Property 3 of proportionalities
9.	^BD/_AD = ^EC/_AE	Segment Addition Postulate
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 ΔADE ~ ΔABC in Figure 14.5. You can prove that ¯DE ‌ ‌ ¯BC.

Example 2: If ΔADE ~ ΔABC as shown in Figure 14.5, prove that ¯DE ‌ ‌ ¯BC.

Figure 14.5ΔADE ~ ΔABC.

Solution: Your game plan is quite simple. Because ΔADE ~ ΔABC, you know that ∠ADE ~= ∠ABC. Because ∠ADE and ∠ABC are congruent corresponding angles, you know that ↔DE ‌ ‌ ↔BC by Theorem 10.7.

	Statements	Reasons
1.	ΔADE ~ ΔABC	Given
2.	∠ADE ~= ∠ABC	Definition of ~
3.	↔DE and ↔BC are two lines cut by a transversal ↔AB	Definition of transversal
4.	∠ADE and ∠ABC are corresponding angles	Definition of corresponding angles
5.	¯DE ‌ ‌ ¯BC	Theorem 10.7

Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.

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See also:

Geometry: The Big Three