![]() Depending on whether or not you think these theorems are things you want to be true in your theory, this could be a reason in support of adopting exclusive definitions. In other words: the theorems are true for the general category only if the more specific category is excluded. perpendicular bisectors) would be concurrent, and the "figure" formed by them would be a single point. rhombus) is a square, then the angle bisectors (resp. However, if inclusive definitions are used, these theorems are false, because if the rectangle (resp. if "rectangle" is defined as "a quadrilateral that is equiangular but not equilateral" and if "rhombus" is defined as "a quadrilateral that is equilateral but not equiangular". These theorems are true statements if rectangle and rhombus are defined exclusively, i.e. The perpendicular bisectors of any rhombus intersect at four points which form the vertices of another rhombus. The angle bisectors of any rectangle intersect at four points which form the vertices of a square. However, consider the following theorems: It's something of a truism to say that inclusive definitions are better, because any properties proved of a more general category automatically apply to all special cases contained within it. The main book cited there will show you that there isn't really any virtue gained by using the exclusive-version, and that there are virtues for using the inclusive-version. I wrote a rather detailed answer with a bibliography at this link, and I think you will be interested in checking it out. I did some research on this in the past, and found many good resources on the topic. (That is not a definitive description: perhaps the problem does pervade other places.) I found in my searches that outside of American K-12 education (i.e., in higher education, and basically any non-American school) the inclusive version was the definition taken for granted. In short, the inclusive-version definition of trapezoids (your second definition) definitely has a stronger case in terms of modern mathematics. To add to the confusion, this paradigm is inconsistently applied, as the same people who use the exclusive-version definition of trapezoid do not usually also declare that rectangles are nonsquare. This framework of two pairs of consecutive congruent sides, opposite angles congruent, and perpendicular diagonals is what allows for the toy kite to fly so well.The idea of having "exclusive definitions" that put types of quadrilaterals into buckets instead of a hierarchy is a holdover from antiquity (even back to Euclid, if my vague memory of what I read is true.) I think its main proponents are often people who learned (and now teach) these mathematical ideas dogmatically, without a full appreciation of 20th century advancement in presentation of mathematics. This means, that because the diagonals intersect at a 90-degree angle, we can use our knowledge of the Pythagorean Theorem to find the missing side lengths of a kite and then, in turn, find the perimeter of this special polygon. ![]() And while the opposite sides are not congruent, the opposite angles formed are congruent. In fact, a kite is a special type of polygon.Ī kite is a quadrilateral that has two pairs of consecutive congruent sides. The way a toy kite is made has everything to do with mathematics! The first thing that pops into everyone’s mind is the toy that flies in the wind at the end of a long string.īut have you ever stopped to wonder why a kite flies so well? Determining if the given quadrilateral is a trapezoid, and if so, is the trapezoid isosceles?.Using these properties of trapezoids to find missing side lengths, angles, and perimeter.In the video below, we’re going to work through several examples including: So if we can prove that the bases are parallel and the diagonals are congruent, then we know the quadrilateral is an isosceles trapezoid, as Cool Math accurately states. But there’s one more distinguishing element regarding an isosceles trapezoid.Ī trapezoid is isosceles if and only if its diagonals are congruent.
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