Skowhegan Area High School
 Parallelograms
 Inductive Proofs, Vertical Angle Theorem, Similar Triangles 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 angles. 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!