More Proofs Using the Parallel Tests (Day 43)

The point of today's lesson is for the students to practice more proof writing. So far, we've only learned a limited number of postulates and theorems that can be used in proofs -- the heavy lifters are to come in the next three chapters. Still, there are a few interesting proofs possible here:

Given: AD and BC intersect at E,
           Angles DCE and DEC have equal measure,
           Angles AEB and ABE have equal measure.
Prove: DC || AB

A student may try drawing a picture of the above before attempting the following proof:

Statements             Reasons
1. mDCE = mDEC 1. Given
2. mDEC = mAEB  2. Vertical Angles Theorem
3. mDCE = mAEB  3. Transitive Property of Equality
4. mAEB = mABE   4. Given
5. mDCE = mABE  5. Transitive Property of Equality
6. DC || AB              6. Alternate Interior Angles Test

Proofs like these prepare students for proofs later on in the text. Notice that the two isosceles triangles DCE and ABE are in fact similar -- and indeed congruent if we were given another statement such as "E is the midpoint of BC" (by ASA). But oftentimes when we reach the triangle congruence theorems, the students forget to check for vertical angles and other reasons that angles may be congruent other than the given information. So this proof lets the students practice thinking about statements and reasons before they are bogged down with theorems to memorize.