Inductive Proofs, Vertical Angle Theorem,
|Let’s look at the above situation through an inductive proof.
Remember, an inductive proof is one where you look at examples and develop a plausible
reason. It may or may not be always true.
Two small cars were driving away from a jackknifing truck. These cars each traveled
an equal distance before smashing into each other. Because the distance and
force was the same for both cars, the two small vehicles bounced off of each
other again with the same angle that they created just before the crash. The
two cars then had to screech on the brakes before they hit the train that passed
in front of them. Interestingly, the path of the train was exactly parallel
to the jackknifed truck that they had left behind. So, with this catastrophic
driving situation, you can actually create two triangles:
|Once the drivers recovered from the shock of the accident, they
noticed that the two triangles, before and after the crash were similar! Angles
5 and 6 are congruent because of the Vertical Angle Theorem.
Because the jackknifed truck and the train provided parallel sides, angles 2 and
3 are alternate interior angles, as well as angles 1 and 4. The drivers have shown
inductively, that the two angles are congruent through AA similarity.
They became even more interested in investigating the accident scene and found
out that the distance between angles 1 and 5 was 80% of the distance between angles
6 and 4. They also discovered that the distance between angles 2 and 5 was 80%
of the distance between angles 6 and 3! Similarity could then be inductively proven
through SAS similarity, using angles 5 and 6 as the included
Their final measurement was made for the distances between angles 1 and 2 and
then 3 and 4. Again, the first measurement was 80% of the second. Similarity could
then be inductively proven through SSS similarity.
What an interesting accident!