Chapter 1 Euclidean Geometry Exercise 1.1. Suppose that at the spring solstice the angle between a star, the Earth, and the sun measures 79.1◦, and at the fall solstice the angle measures 100.8◦. How far away is the star from the Earth? (The Earth is 93 million miles from the sun.) The solution is in the text. Exercise 1.2. The numbers in Exercise 1.1 were cooked up. From your general knowledge, are these numbers reasonable? Explain. Solution. In Exercise 1.1, we found that the star is 100,000,000,000 miles away. Let us convert this figure to light years. A light year is how far light travels in a year. Since light travels at a speed of 186,000 miles per second, one light year is the same as 1 lightyear = 186, 000 miles sec yr 3600 sec hr 24 hrs day 365 day yr = 5.87 × 1012 miles. Thus, the distance to the star in Exercise 1.1 is 1011 miles 1 lightyear 5.87 × 1012 miles = .017 lightyear. Since this is much smaller than 4.3 lightyears, the distance to the nearest star (other than the Sun), we conclude that the numbers are not reasonable. Exercise 1.3. What order of magnitude is the difference between 180◦ and the sum of the angles for