• Journal of Educational Research in Mathematics
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• v.15 no.2
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• pp.131-145
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• 2005

This study intends to analyze didactically on triangle-determining conditions and triangle-congruence conditions. The result of this study revealed the followings: Firstly, many pre-service mathematics teachers and secondary school students have insufficient understanding or misunderstanding on triangle-determining conditions and triangle-congruence conditions. Secondly, the term segment instead of edge may show well the concern of triangle-determining conditions. Thirdly, when students learn the method of finding six elements of triangle using the law of sines and cosines in high school, they should be given the opportunity to reflect the relation and the difference between triangle-determining situation and the situation of finding six elements of triangle. Fourthly, accepting some conditions like SSA-obtuse as a triangle-determining condition or not is not just a logical problem. It depends on the specific contexts investigating triangle-determining conditions. Fifthly, textbooks and classroom teaching need to guide students to discover triangle-deter-mining conditions in the process of inquiry from SSS, SSA, SAS, SAA, ASS, ASA, AAS, AAA to SSS, SAS, ASA, SAA. Sixthly, it is necessary to have students know the significance of 'correspondence' in congruence conditions. Finally, there are some problems of using the term 'correspondent' in describing triangle-congruence conditions.

• School Mathematics
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• v.16 no.2
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• pp.219-236
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• 2014

We recognized that most teachers are having insufficient understanding or misunderstanding about congruent conditions of triangles. So the purpose of this study was to analyze teachers's understanding about congruent conditions of triangles and to find the causes of teachers's misunderstanding. Most teachers have been misunderstanding that triangle determining- conditions are only 3 ways(SSS, SAS, ASA). And they have wrong confidence that 2 sides and a non included angle(ASS) is not always able to make one triangle. This study found that these teachers's misconception was from the textbook using now. As the result of this study, we suggested 7 improvement ways about planning of curriculum, writing of textbook and teacher training course.

• The Mathematical Education
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• v.49 no.1
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• pp.53-65
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• 2010

The congruent conditions of triangles' plays an important role to connect intuitive geometry with deductive geometry in school mathematics. It is induced by 'three determining conditions of triangles' which is justified by classical geometric construction. In this paper, we analyze the essential meaning and geometric position of 'congruent conditions of triangles in Euclidean Geometry and investigate introducing processes for them in the Elements of Euclid, Hilbert congruent axioms, Russian textbook and Korean textbook, respectively. Also, we give justifications of construction methods for triangle having three segments with fixed lengths and angle equivalent to given angle suggested in Korean textbooks, are discussed, which can be directly applicable to teaching geometric construction meaningfully.

• International Journal of Ocean System Engineering
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• v.1 no.2
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• pp.95-101
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• 2011

Perforated plates with cutouts (or holes) are widely used in structural members. These cutouts provide stress concentration in plates. Extensive studies have been carried out on stress concentration in perforated plates, which consider cutout shapes, boundary conditions, bluntness of cutouts, and more. This study presents stress concentration analyses of perforated plates with not only various cutouts and bluntness but also different cutout orientations. Especially, the effect of cutout orientation on stress concentration is emphasized since structural members have become more complicated recently. To obtain stress concentration patterns, a finite element program, ANSYS, is used. For the designated goal, three parameters are considered as follows: the shapes of polygonal cutouts (circle, triangle, and square), bluntness (a counter measure of radius ratio, r/R), and rotation of cutouts (${\theta}$). From the analyses, it is shown that, in general, as bluntness increases, the stress concentration increases, regardless of the shape and rotation. A more important finding is that the stress concentration increases as the cutouts become more oriented from the baseline, which is the positive horizontal axis (+x). This fact demonstrates that the orientation is also a relatively significant design factor to reduce stress concentration. In detail, in the case of the triangle cutout, orienting one side of the triangle cutout to be perpendicular to the applied tensile forces is preferable. Similarly, in the case of the square cutout, it is more advantageous to orient two sides of square cutout to be perpendicular to the applied tensile force. Therefore, at the design stage, determining the direction of a major tensile force is required. Then, by aligning those polygon cutouts properly, we can reduce stress concentration.