Reading for Week 3

B&B, Chapter 4.

This chapter contains the basic properties of parallel lines and transversals. It also introduces coordinates and the slope of a line.

Assignment 4 (50 points, Due Mon 10/16)

In the problems from B&B, you can use whatever was proved that occurs in the book previous to the problems (this also includes homework you have done prior to the problem). Otherwise, use principles 1-16.

4.1 Altitudes (10 points)

B&B, p. 75 - #10,11,12. Suggestion: Read over all 3 problems and think about them before you begin. Can you combine arguments to cover all cases in one proof? Do you have to do more geometry for 12 or just use logic?

4.2 Distances in a cube (5 points)

B &B p. 101, #12, 13, 14, 15

4.3 Constructing parallels (5 points)

Draw a line AB and a point C not on the line. Construct with straightedge and compass the line through C that is parallel to line AB.

4.4 Parallelograms (10 points)

Prove the following problems on B&B, pp. 113-114. Important: Use the definition of a parallelogram at the beginning of the Exercises on p. 113. Do not use other equivalent definitions. For tools, you can use anything proved in B&B through p. 122.

  1. Prove 2 and its converse 5.
  2. Prove 3 and its converse 7.
  3. Prove 6.
  4. A rhombus is defined to be a quadrilateral with four equal sides. Prove that a rhombus is a parallelogram

4.5 A special triangle (10 points)

Let ABC be an isosceles triangle (AB = AC). Suppose there is a point D on segment AB so that CD = CB. Let us label the lengths: AB = AC = x and CD = CB = y.

(a) With these assumptions what is length BD in terms of x and y? Also, write the length AD in terms of x and y.

(b) If we make the additional assumption that triangle DAC is also isosceles (i.e., DA=DC), find the ratio x/y. The answer should be a real number (no unknowns).

4.6 Special Midpoint Quadrilaterals (10 points)

(a) Consider this statement: If the midpoint quadrilateral of quadrilateral ABCD is a rectangle, then ABCD is a kite. Prove it if it is true. If it is false, give a counterexample and, if you can, explain for which quadrilaterals the midpoint quadrilateral is a rectangle.

(b) Consider this statement: If the midpoint quadrilateral of ABCD is a rhombus, then ABCD is a rectangle. Prove it if it is true. If it is false, give a counterexample and, if you can explain for which quadrilaterals the midpoint quadrilateral is a rhombus.

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