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Homework #4
Please begin each problem on a new page. Save your finished assignme
Homework #4
Please begin each problem on a new page. Save your finished assignment as a single pdf and submit to Canvas. Write your own proof of the Saccheri-Legendre theorem. Include a statement of the theorem, what we want to show, what is given, and an appropriate labeled diagram. Your proof should follow the presentation in class and in Wolfe (pg. 35) but should be written formally in your own words. The following statements are equivalent to the Euclidean parallel postulate. Explain in your own words why the EPP, or something else equivalent, is needed for that statement to hold. Include relevant diagrams. Note: you are not being asked to formally deduce one statement from another, but instead to explain why the EPP is logically equivalent to each statement.
Similar non-congruent triangles exist.
The Pythagorean theorem holds for right triangles.
A rectangle exists. The following statements are true in Euclidean geometry. Rewrite the underlined part of the statement so that each one is a valid statement in neutral geometry. If the statement is already valid in neutral geometry, say so and briefly explain how you know.
Example: The sum of the measure of angles in a triangle is exactly 180 degrees.
In neutral geometry, the sum of the measure of angles in a triangle is less than or equal to 180 degrees.
Example: The exterior angle of a triangle is greater than either remote interior angle. This is already a valid statement in neutral geometry. If l is any line and P is any point not on l, there exists exactly one line m through P parallel to l. If l is any line and P is any point not on l, there is exactly one line through P perpendicular to l.
The summit angles of a Saccheri quadrilateral are congruent and equal to 90 degrees.
If two lines cut by a transversal have a pair of congruent alternate interior angles, then the two lines are parallel.
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