In my book Some Adventures in Euclidean Geometry book (free download), first published in 1994, I suggested the name 'angle quadrilateral' for a general quadrilateral having at least one pair of opposite angles equal. However, I would now instead like to propose the more appropriate name 'opposite angle quadrilateral'.
Note that this quadrilateral is the dual or analogue of an Opposite Side Quadrilateral. Perhaps somewhat unexpectedly, an 'opposite angle quadrilateral' has an interesting property as shown below.
Another Property of an Opposite Side Quadrilateral
Given an opposite angle quadrilateral ABCD with ∠BAD = ∠BCD and label the intersections of the extensions of BA and CD, and of AD and BC, respectively as P and Q. Then the angle bisectors of ∠BPC, ∠AQB, ∠ABC and ∠ADC form an isosceles trapezium LNOK as shown below.
Illustration
The dynamic sketch below illustrates the above property with some measurements.
1) Explore the result dynamically by dragging any of A, C, D or X.
2) Do the results always hold when ABCD becomes concave or crossed? Explore!
3) What happens when AB // CD or AD // BC? Do the above results still hold?
A Property of an Opposite Angle Quadrilateral
Challenge
5) Can you explain why (prove that) the above property of an opposite angle quadrilateral is true?
6) Compare your proof with the proof on pp. 97-98 of my Some Adventures in Euclidean Geometry (free download) book.
Further Challenges
7) Prove that an opposite angle quadrilateral that has a pair of adjacent sides equal (but the pair of adjacent sides does not include one of the pair of equal opposite angles) is a kite.
8) Prove that an opposite angle quadrilateral with perpendicular diagonals is a kite.
9) Prove that an opposite angle quadrilateral that is tangential (convex or concave) is a kite.
10) Prove or disprove the following statement: An opposite angle quadrilateral with one diagonal bisecting the other is a kite.
11) Prove or disprove the following statement: An opposite angle quadrilateral with a pair of opposite sides equal is a parallelogram.
12) Proofs of 7), 8) and 9) are respectively given on p. 120, p. 121, and p. 174, and compare your answers to 10) and 11) respectively to p. 122 and pp. 112-113; 128-129 in my Some Adventures in Euclidean Geometry (free download) book.
Special Case
An interesting special case of the theorem at the top is that triangles KLM and MNO become equilateral when ∠BAD = ∠BCD = 60o.
References
De Villiers, M. (2009). Some Adventures in Euclidean Geometry book (free download), Lulu Publishers: Dynamic Mathematics Learning.
Some Related Links
Opposite Side Quadrilateral Properties by Kalogerakis
Another Property of an Opposite Side Quadrilateral
Quadrilateral Similar Triangles Collinearity ((A generalization of a theorem for 'equilic' quadrilaterals)
A Hierarchical Classification of Quadrilaterals
Definitions and some Properties of Quadrilaterals
Van Aubel's Theorem and some Generalizations
A diagonal property of a Rhombus constructed from a Rectangle
The 120o Rhombus (or Conjoined Twin Equilateral Triangles) Theorem
Matric Exam Geometry Problem - 1949 (A variation of Reim's theorem)
Golden Rhombus
A Rhombus Angle Trisection Result
A problem by Paul Yiu and its generalization
More Properties of a Bisect-diagonal Quadrilateral
A Van Aubel like property of an Orthodiagonal Quadrilateral
A Van Aubel like property of an Equidiagonal Quadrilateral
SA Mathematics Olympiad Problem 2016, Round 1, Question 20
SA Mathematics Olympiad 2022, Round 2, Q25
An extension of the IMO 2014 Problem 4
A 1999 British Mathematics Olympiad Problem and its dual
Pitot's theorem about a Tangential Quadrilateral
Extangential Quadrilateral
Theorem of Gusić & Mladinić
Constructing a general Bicentric Quadrilateral
Conway's Circle Theorem as special case of Side Divider (Windscreen Wiper) Theorem
Japanese Circumscribed Quadrilateral Theorem
Some External Related Links
Quadrilateral (Wikipedia)
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Created with WebSketchpad by Michael de Villiers, 12 Dec 2024; updated 17 Dec 2024.