Assignment 3 (Due Wednesday, 10/13)

3.1 Angle sum of a polygon

1. For a convex n-sided polygon, what is the sum of the interior angles.
2. Prove your statement above.
3. Is this formula still true for a nonconvex quadrilateral? Justify your answer..

3.2 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.

3.3 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.)

3.4 Tangent circles

1. Given an angle ABC, prove that a circle is tangent to rays BA and BC if and only if the center P of the circle is on the angle bisector of ABC.
2. Prove that the bisectors of the angles of any triangle ABC are concurrent.
3. Draw a (general) triangle and construct the circle inscribed in the triangle. Explain the key steps of the construction and tell why the construction works.

3.5 Inscribed circles

1. The lines AB and CD are parallel.  Construct all circles tangent to all 3 lines in the figure.
2. Explain carefully the steps of your construction and why it works.

Given a quadrilateral ABCD, prove that there is exactly one circle tangent to the rays AD, BC and segment AB. Tell how to construct this circle and and give an example