13.1 Angles and Parallel Lines
If two lines are parallel and a line is drawn across them that line is called a transversal. In the diagram below the two black lines are parallel and the red line is the transversal. The letters mark different angles. Which angles do you think are equal to angle A of the angles B C and D? What about the angles E F G and H. Are any of them equal to angle A?
Angle A is equal to angle D. Angle A is also equal to angle E. Angle E is equal to angle H. Therefore Angle A is also equal to angle H. Angles D and E are called alternate interior angles because they are interior to the two parallel lines but on opposite sites of the transversal. Alternate interior angles alway equal each other when the lines are parallel. Angles C and F are also alternate interior angles and they also equal each other.
These relationships between angles were used 3000 years ago by a Greek mathematician named Erastosthenes to calculate the size of the earth. He knew that the suns rays were parallel. In the diagram below he knew that angles A and B were alternate itnerior angles and therefore the same. He was able to measure angle A. Once he knew angle B he could calcualte how many angle Bs made a full circle around the world. A full circle is 360 degrees. Angle A was about 7.2 degrees so angle B was also 7.2 degrees. The number of angle Bs that would make a full circle is 360 degrees/7.2 degrees = 50. So if you know the distance between Alexandria and Syene you can multiply by 50 to get the circumference of the earth which is what Erastosthenes did.
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