I have three questions about a quadrilateral with the following properties: It is convex. It has exactly one pair of congruent opposite sides. It has exactly one pair of congruent opposite angles. It is not a parallelogram. Does such a quadrilateral exist? Is it possible to construct such a quadrilateral with compass and straightedge? If the construction is not possible, why not?

Matonya

Matonya

Answered question

2022-08-10

I have three questions about a quadrilateral with the following properties:

1. It is convex.
2. It has exactly one pair of congruent opposite sides.
3. It has exactly one pair of congruent opposite angles.
4. It is not a parallelogram.

Does such a quadrilateral exist? Is it possible to construct such a quadrilateral with compass and straightedge? If the construction is not possible, why not?

Answer & Explanation

pelvogp

pelvogp

Beginner2022-08-11Added 18 answers

Yes, such quadrilaterals exist -- even constructible ones.

For example, the quadrilateral ABCD with
D A B = 60 A B D = 45 A D B = 75 B C D = 60 C B D = 105 C D B = 15
satisfies the requiblack conditions.
grippeb9

grippeb9

Beginner2022-08-12Added 2 answers

Let ABC be an isosceles right triangle with AC = BC. On base AB make BD = AC and join CD. Thus angle BCD = angle BDC = 67.5 degrees. On CD, congruent to triangle CDB, construct triangle CDB' in a plane perpendicular to that of triangle ACD, forming quadrilateral ACB'D. Since angle B'CA is less than 90 degrees (it's less than the angle AC makes with a perpendicular to the plane of ACD at A), while angle B'DA is greater than 90 degrees (it's greater than the angle AD makes with a perpendicular to ACD at D), these angles are not equal. But angle CAD = opposite angle CB'D = 45 degrees. And AC = opposite side B'D. But AC is not parallel to B'D, since they are not co-planar, and for the same reason CB' is not parallel to AD. Therefore, a quadrilateral ACB'D has been constructed (with straightedge and compass) in which only one pair of opposite sides and angles are equal, and which is not a parallelogram.

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