Assignment 4 (Due Wed. 10/19)

Reading and studying. You have now had Chapters 1, 2, 3 and 4 in BB as a reading assignment. This is a serious assignment. It will help you do the assignment problems and succeed on the tests. Among other things.

• Learn carefully and thoroughly the (correct and complete) statements of definitions and the theorems (these are handily summarized at the end of each chapter).
• Look over the proofs of the theorems, check which ones were emphasized in class or lab. You should know how to prove these key statements.
• Read over the starred problems. Know them as "facts". Be prepared to prove most of them. You have done quite a few on assignments or in class or in lab.

Problem 4.1 Special parallelograms

1. Prove that a parallelogram is a rectangle if and only if the diagonals are congruent.
2. Prove that a parallelogram is a rhombus if and only if the diagonals are perpendicular.

Problem 4.2 Strip figure

1. Draw a quadrilateral using an ordinary ruler in this way. Place the ruler on a sheet of paper and draw the two parallel lines formed by the two edges of the ruler. Now rotate the ruler and draw two more parallel lines the same way. The four lines will intersect at 4 points forming a quadrilateral ABCD.
2. Prove that the quadrilateral ABCD is a rhombus. (More formally, if ABCD is a parallelogram so that the distance between lines AB and CD is equal to the distance between the lines AC and BD, then ABCD is a rhombus.)

Problem 4.3 Some ratios

1. Prove BB problem 26, on page 64.
2. Solve BB problem 28. Show reasoning.

Problem 4.3 Some ratios

1. Prove BB problem 26, on page 64.
2. Solve BB problem 28. Show reasoning.

Problem 4.4 Altitudes

1. Prove BB problems 10-11, on page 75. You can consider this one problem with 3 cases: C acute angle, C right angle, C obtuse angle.

Problem 4.5 Parallels in a triangle

1. Construct a figure like BB p. 115, figure 11. Prove BB problems 21-22, on page 115. You can combine this into one problem.
2. In the figure of triangle ABC, D, E, F are midpoints of AB, BC, AC. Let G be the intersection of AE and CD. Find the ratios AG/AE and CG/CD (the answer should be a number).
3. Continuing with b, let H be the intersection of BF and AE. Find the ratios AH/AE and BH/DF. What can you conclude about the relationship between G and H?
4. The line segments AE, CD, BF connecting vertices to midpoints are called the medians of ABC. State clearly an interesting fact (or facts) about the intersections of the 3 medians that you can conclude from earlier parts of this problem.

Problem 4.6 Trapezoids

1. Prove BB problem 23, page 116. Explain clearly what the statement of the problem means.
2. Prove BB problem 24, page 116.
3. In a trapezoid, prove that the intersection point of the diagonals and the midpoints of the two parallel sides are all collinear.