Class Conjectures, Spring 2017

These are the conjectures and questions proposed by the class this term, ordered by date.

January 13

Conjecture A (Pace): Let ABCD be a rhombus. Then angle BAC and BDC taken together make a right angle.

Conjecture B (Barker-Lewis): Let ABCD be a rhombus. If angle BAC is congruent to angle BDC, then ABCD is a square.

Conjecture C (class): It is possible to construct a rhombus which is not a square?

January 18

Conjecture D (Reihman): Suppose that we have a diagram where A and D lie on the same side of line BC. Then angles ABC and BCD taken together make two right angles if and only if lines AB and CD are parallel.

January 27

Question E: Is it possible to construct a kite which has no equilateral triangle? (The Van Donselaar examples all have triangle ABC as an equilateral triangle.)

Question F: What is an “interior angle?”

Question G: What is the “interior” of a polygon?

Question H: How do we differentiate between the “normal-looking” and the “odd-looking” kites? Is there some geometrical way of distinguishing these?

January 30

Conjecture I: (Lewis) Given a segment AB and an angle XYZ, it is possible to construct a rhombus ABCD with angle ABD congruent to angle XYZ using a compass and straightedge.

February 1

Question J: (Peters) Can we construct a kite with a given pair of sides and a given pair of opposite angles. More specifically: Given angles XYZ and \(\alpha \beta \gamma\), and segments TJ and PQ, is it possible to construct a kite ABCD such that AB is congruent to TJ, BC is congruent to PQ, angle DAB is congruent to angle XYZ and angle DCB is congruent to angle \(\alpha \beta \gamma\)?

Lemma K: (Schmidt) Given segments AB and XY, one can construct triangle ABC so that BA and BC are both congruent to XY.