Draw a quadrilateral with each characteristic that is not a parallelogram. 11. two pairs of congruent angles 12. two pairs of congruent sides 13. one pair of congruent opposite angles 14. one pair of opposite sides parallel and the other pair congruent 15. The legs of the adjustable keyboard stand are connected at their midpoints. Explain why the keyboard remains level to the floor as the stand is raised and lowered.

Below are one set of acceptable answers. (Any correct example is acceptable.) In each case you must “draw” (or describe) a quadrilateral that has the given property but is not a parallelogram. For item 15 you must explain why the connecting bar stays horizontal. ────────────────────────────── 1. Two pairs of congruent angles One acceptable answer is an isosceles trapezoid. (An isosceles trapezoid has exactly one pair of parallel sides so it is not a parallelogram. In an isosceles trapezoid the base angles (the angles adjacent to each base) are congruent in pairs; that is, the two angles at the long base are congruent and the two at the short base are congruent.) For example, label the vertices A, B, C, D in order. Draw AB (the longer base) horizontal at the bottom and CD (the shorter base) horizontal near the top so that AB ∥ CD. Choose the sides AD and BC so they are congruent (and not parallel) but not parallel to AB and CD. (See the sketch below.) D___________C / \ / \ A /_____________\ B Here ∠A = ∠B and ∠C = ∠D. ────────────────────────────── 2. Two pairs of congruent sides A common example is a kite. In a kite the two pairs of adjacent sides (say, sides AB = AD and sides CB = CD) are congruent. (In general a kite is not a parallelogram because its opposite sides are not parallel.) Sketch a kite as follows: C /\ / \ D /____\ B \ / \/ A Here the two sides meeting at A and the two sides meeting at B are each congruent. ────────────────────────────── 3. One pair of congruent opposite angles Now we need a quadrilateral in which exactly one pair of opposite angles is congruent. One acceptable example is a kite that is not symmetric about both diagonals. (Recall that in a typical kite the angles between the two pairs of congruent adjacent sides are not equal except for the pair at the “non‐tip” vertices.) For example, if you let the angles at vertices B and D be different and only the two opposite angles at, say, A and C are equal, then the quadrilateral has one pair of congruent opposite angles. A possible drawing (with vertices in order A, B, C, D): B / \ / \ A / \ C \ / \ / \ / D Choose the lengths so that only ∠A and ∠C are equal while ∠B and ∠D are not. ────────────────────────────── 4. One pair of opposite sides parallel and the other pair congruent A classic answer is an isosceles trapezoid. In an isosceles trapezoid one pair of opposite sides (the bases) are parallel, and the legs (the non‐parallel sides) are congruent. (Such a quadrilateral is not a parallelogram because only one pair of sides is parallel.) See the drawing under item 1 above. ────────────────────────────── 5. Keyboard stand problem The legs of an adjustable keyboard stand are connected at their midpoints. Why does the keyboard remain level as the stand is raised and lowered? Answer: Because the connecting bar joins the midpoints of the legs, it is a midsegment of the (implied) triangles formed by the legs and the floor. According to the Midsegment Theorem, the segment joining the midpoints of two sides of a triangle is parallel to the third side. Here that means the bar is always parallel to the floor. Therefore, no matter how the legs are repositioned when the stand is raised or lowered, the bar (and hence the keyboard attached to it) remains horizontal. ────────────────────────────── Any answers that present these examples (or others logically equivalent to these) are correct.