[pt] Neste trabalho, o foco principal é o Teorema de Morley, cuja formulação tem como base um dos três problemas clássicos da Geometria: a trissecção de um ângulo. A partir da contextualização histórica, procura-se inserir o tema como motivação ao estudo da Geometria. Seja pela riqueza dos conteúdos envolvidos ou pela dificuldade na sua construção, o Triângulo de Morley é um belo exemplo de aplicação a ser trabalhado em sala de aula. Em paralelo é feita uma análise crítica das dificuldades encontradas no processo de ensino e aprendizagem em Matemática, principalmente em Geometria, visando oferecer subsídios importantes à atividade docente e impactando positivamente o seu trabalho. Com as demonstrações e aplicações apresentadas, procura-se sedimentar conhecimentos adquiridos, bem como apontar caminhos para soluções de diversos problemas geométricos similares. / [en] In this work, the main focus is the Morleys Theorem, whose formulation is based on one of three classical problems of Geometry: a trisection of an angle. From the historical context, it tries to insert the subject as motivation to the study of geometry. By the richness of the content or the difficulty involved in its construction, the Morleys triangle is a fine example of application to work with in the classroom. In parallel, a critical analysis of the difficulties that are found on the process of teaching and learning Mathematics, especially Geometry, is done in order to offer important benefits to the teaching activity and positively impacting their work. With demonstrations and applications that are shown it tries to fix up the acquired knowledge, as well as identifying ways to solutions of several similar geometric problems.

Neste trabalho fazemos um estudo detalhado sobre o tema Trigonometria. A trigonometria é um tema bastante discutido em sala de aula durante o ensino médio. Não apenas apresentamos resultados sobre o tema mas também suas provas e justificativas, assim como exemplos e exercícios com o objetivo de ter um material completo para professores do ensino médio que desejem estudar tais tópicos. Em seguida apresentamos algumas aplicações da Trigonometria que podemos encontrar em nosso dia-a-dia, também aqui o objetivo é apresentar motivação para o estudo deste importante assunto e tão frequente nos vestibulares atualmente. Finalmente, apresentamos uma atividade realizada com meus alunos em sala de aula. Esta dissertação foi desenvolvida como parte dos requisitos necessários para a obtenção do título de mestrado acadêmico junto ao Instituto de Ciências Matemáticas e de Computação (ICMC), da Universidade de São Paulo (USP). / In this dissertation we present a detailed study about Trigonometry. This subject is frequently discussed em classes during High school courses. We do not only present the main results about Trigonometry but also their proofs, as well examples and exercises. Our main objective here is obtain a complete text for high school teachers. We also present some applications of Trigonometry that can be easily find in our life. Here our main objective is to motivate the study of this important subject that appears so frequently in the exams for universities entrance. To conclude, we present an activity realized with high school students. This dissertation was developed as part of the requirements necessary for the obtension of the degree of Mathematics Professional Master at Instituto de Ciências Matemáticas e de Computação da Universidade de São Paulo (ICMC-USP).

Triangle-mesh modeling, as one of the approaches for representing images based on nonuniform sampling, has become quite popular and beneficial in many applications. In this thesis, image representation using triangle-mesh models and its application in image scaling are studied. Consequently, two new methods, namely, the SEMMG and MIS methods are proposed, where each solves a different problem. In particular, the SEMMG method is proposed to address the problem of image representation by producing effective mesh models that are used for representing grayscale images, by minimizing squared error. The MIS method is proposed to address the image-scaling problem for grayscale images that are approximately piecewise-smooth, using triangle-mesh models. The SEMMG method, which is proposed for addressing the mesh-generation problem, is developed based on an earlier work, which uses a greedy-point-insertion (GPI) approach to generate a mesh model with explicit representation of discontinuities (ERD). After in-depth analyses of two existing methods for generating the ERD models, several weaknesses are identified and specifically addressed to improve the quality of the generated models, leading to the proposal of the SEMMG method. The performance of the SEMMG method is then evaluated by comparing the quality of the meshes it produces with those obtained by eight other competing methods, namely, the error-diffusion (ED) method of Yang, the modified Garland-Heckbert (MGH) method, the ERDED and ERDGPI methods of Tu and Adams, the Garcia-Vintimilla-Sappa (GVS) method, the hybrid wavelet triangulation (HWT) method of Phichet, the binary space partition (BSP) method of Sarkis, and the adaptive triangular meshes (ATM) method of Liu. For this evaluation, the error between the original and reconstructed images, obtained from each method under comparison, is measured in terms of the PSNR. Moreover, in the case of the competing methods whose implementations are available, the subjective quality is compared in addition to the PSNR. Evaluation results show that the reconstructed images obtained from the SEMMG method are better than those obtained by the competing methods in terms of both PSNR and subjective quality. More specifically, in the case of the methods with implementations, the results collected from 350 test cases show that the SEMMG method outperforms the ED, MGH, ERDED, and ERDGPI schemes in approximately 100%, 89%, 99%, and 85% of cases, respectively. Moreover, in the case of the methods without implementations, we show that the PSNR of the reconstructed images produced by the SEMMG method are on average 3.85, 0.75, 2, and 1.10 dB higher than those obtained by the GVS, HWT, BSP, and ATM methods, respectively. Furthermore, for a given PSNR, the SEMMG method is shown to produce much smaller meshes compared to those obtained by the GVS and BSP methods, with approximately 65% to 80% fewer vertices and 10% to 60% fewer triangles, respectively. Therefore, the SEMMG method is shown to be capable of producing triangular meshes of higher quality and smaller sizes (i.e., number of vertices or triangles) which can be effectively used for image representation. Besides the superior image approximations achieved with the SEMMG method, this work also makes contributions by addressing the problem of image scaling. For this purpose, the application of triangle-mesh mesh models in image scaling is studied. Some of the mesh-based image-scaling approaches proposed to date employ mesh models that are associated with an approximating function that is continuous everywhere, which inevitably yields edge blurring in the process of image scaling. Moreover, other mesh-based image-scaling approaches that employ approximating functions with discontinuities are often based on mesh simplification where the method starts with an extremely large initial mesh, leading to a very slow mesh generation with high memory cost. In this thesis, however, we propose a new mesh-based image-scaling (MIS) method which firstly employs an approximating function with selected discontinuities to better maintain the sharpness at the edges. Secondly, unlike most of the other discontinuity-preserving mesh-based methods, the proposed MIS method is not based on mesh simplification. Instead, our MIS method employs a mesh-refinement scheme, where it starts from a very simple mesh and iteratively refines the mesh to reach a desirable size. For developing the MIS method, the performance of our SEMMG method, which is proposed for image representation, is examined in the application of image scaling. Although the SEMMG method is not designed for solving the problem of image scaling, examining its performance in this application helps to better understand potential shortcomings of using a mesh generator in image scaling. Through this examination, several shortcomings are found and different techniques are devised to address them. By applying these techniques, a new effective mesh-generation method called MISMG is developed that can be used for image scaling. The MISMG method is then combined with a scaling transformation and a subdivision-based model-rasterization algorithm, yielding the proposed MIS method for scaling grayscale images that are approximately piecewise-smooth. The performance of our MIS method is then evaluated by comparing the quality of the scaled images it produces with those obtained from five well-known raster-based methods, namely, bilinear interpolation, bicubic interpolation of Keys, the directional cubic convolution interpolation (DCCI) method of Zhou et al., the new edge-directed image interpolation (NEDI) method of Li and Orchard, and the recent method of super-resolution using convolutional neural networks (SRCNN) by Dong et al.. Since our main goal is to produce scaled images of higher subjective quality with the least amount of edge blurring, the quality of the scaled images are first compared through a subjective evaluation followed by some objective evaluations. The results of the subjective evaluation show that the proposed MIS method was ranked best overall in almost 67\% of the cases, with the best average rank of 2 out of 6, among 380 collected rankings with 20 images and 19 participants. Moreover, visual inspections on the scaled images obtained with different methods show that the proposed MIS method produces scaled images of better quality with more accurate and sharper edges. Furthermore, in the case of the mesh-based image-scaling methods, where no implementation is available, the MIS method is conceptually compared, using theoretical analysis, to two mesh-based methods, namely, the subdivision-based image-representation (SBIR) method of Liao et al. and the curvilinear feature driven image-representation (CFDIR) method of Zhou et al.. / Graduate

O principal objetivo deste trabalho é realizar uma leitura atenta que demonstre a importância do ciúme para a tessitura narrativa nos romances de Italo Svevo: Una vita (1892), Senilità (1898) e La coscienza di Zeno (1923). A partir da ordem de publicação de cada obra, pretende-se desvelar as relações amorosas, desnudando o modo como o ciúme apresenta-se nas histórias vividas pelos personagens. As diversas teorias que discorrem sobre o ciúme, ainda que de modos diferentes, convergem em um ponto, ou seja, no caráter triplo desse sentimento que terá, portanto, o triângulo como a forma geométrica que melhor o representa. Todavia, o ciúme será analisado neste estudo, sobretudo, à luz da teoria do filósofo francês, René Girard. Este trabalho ainda trata de duas questões fundamentais na obra de Svevo, intrinsicamente ligadas ao ciúme: o ambiente triestino, onde se desenrolam as histórias e o caráter inapto de seus protagonistas. / The main goal of this work is to produce an attentive reading that demonstrates the importance of jealousy shaping the narrative web in Italo Svevo\'s novels: Una vita (1892), Senilità (1898) e La coscienza di Zeno (1923). Following the chronological order of each piece, the intent is to unveil the amorous relationships by uncovering the manner jealousy is presented in characters\' histories. The diverse theories lecturing on jealousy -though may diverge in some aspects- end up converging in one: the triple aspect of the feeling, the triangle for its geometrical shape that very well represents it. However, jealousy will be taken in this study, most of all, enlightened by the french philosopher theory of René Girard. Furthermore, this thesis also deals with two important matters of Svevo\'s work, intrinsically linked to jealousy: the Triestian enviroment, where main characters\' storyline and inaptitude flourish.

Levando em consideração a influência das variáveis meteorológicas no desenvolvimento dos cultivos e na definição do rendimento de grãos, o monitoramento da resposta da cultura da soja frente às condições ambientais se torna necessário devido sua importância tanto no cenário internacional quanto nacional. Sabe-se que para a cultura da soja, a estiagem é a causa mais frequente das perdas em safras no Rio Grande do Sul. Neste contexto, testar ferramentas de monitoramento em tempo quase real das condições hídricas das lavouras de soja ao longo do ciclo torna-se relevante e permite verificar se a cultura está em adequadas condições de crescimento e desenvolvimento. Os objetivos deste estudo foram avaliar a aptidão do índice TVDI em representar as condições de disponibilidade hídrica em áreas de cultivo de soja no Rio Grande do Sul; caracterizar e analisar a sensibilidade dos parâmetros da relação entre a temperatura de superfície (TS) e o índice de vegetação por diferença normalizada (NDVI) e, por fim, comparar as abordagens por cena e por safra no processo de parametrização do índice TVDI para a cultura da soja. Para tanto, foram utilizados os produtos MODIS TS (MOD11A2) e NDVI (MOD13A2) com resolução espacial de 1.000 m e temporal de 8 e 16 dias, respectivamente, assim como dados de precipitação pluvial de 3 estações meteorológicas localizadas na área de estudo e dados oficiais do IBGE de área e de rendimento. Os resultados mostraram que a abordagem de parametrização por cena possibilita o entendimento da distribuição espacial das condições hídricas da área de estudo frente aos limites hídricos de cada imagem. Os parâmetros "b" e “TSmin” utilizados para determinar as retas dos limites seco e úmido do triângulo evaporativo na determinação do TVDI com parametrização por cena podem auxiliar na compreensão do perfil temporal deste índice ao longo da safra. Já a parametrização por safra, mostrou aptidão do índice em representar o momento e a frequência de ocorrência de restrição hídrica ao longo do ciclo da cultura. O método do triângulo evaporativo para a obtenção do índice TVDI mostra-se eficiente, pois permite a partir de dados de sensoriamento remoto, o entendimento do padrão espacial e temporal de resposta da vegetação frente à disponibilidade hídrica, além de informações de grande utilidade para sistemas de monitoramento e estimativas de safras. / In view of the influence of meteorological variables on crop growth and grain yield, monitoring the response of soybean to different environmental conditions is essential, given the international and domestic importance of this crop. Drought is the leading cause of soybean crop losses in the Brazilian state of Rio Grande do Sul. In this context, testing tools for monitoring in near real time the water conditions of soybean fields throughout their cycle becomes relevant and allows for checking if the crop is under proper conditions for growth and development. The objectives of this study were: to assess the adequacy of the temperature–vegetation dryness index (TVDI) to represent water availability status in soybean growing areas in Rio Grande do Sul; to characterize and analyze the sensitivity of the surface temperature (TS)/normalized difference vegetation index (NDVI) ratio; and to compare the scene-specific and crop-specific approaches for parameterization of TVDI. To this end, the MODIS TS (MOD11A2) and NDVI (MOD13A2) products were used, with a spatial resolution of 1000 m and a temporal resolution of 8 and 16 days respectively, as well as precipitation data from 3 weather stations located in the study area and official area and yield data from the Brazilian Institute of Geography and Statistics (IBGE). Results showed that the scene-specific parameterization approach was able to provide an understanding of the spatial distribution of water status in the area of interest as derived from the dry and wet edges of each image. With scene-based parameterization, the “b” and “TSmin” parameters used to determine the dry and wet edges of the TS/NDVI can help elucidate the temporal profile of this index throughout the harvest period. With crop-type parameterization, the TVDI was able to represent the timing and frequency of limited water availability during the crop cycle. The triangle method proved effective in deriving the TVDI, as it provides a means of using remote sensing data to understand the spatiotemporal response pattern of vegetation to water availability, as well as other information that can be highly useful for monitoring systems and crop estimation.

Casos de fraudes têm ocorrido, freqüentemente, no mercado mundial. Diversos são os profissionais envolvidos nestes casos, inclusive os da classe contábil. Os escândalos contábeis, especialmente os mais famosos, como os incididos nas empresas Enron e WordCom, acenderam para uma maior preocupação em relação à conduta ética dos profissionais de contabilidade. Como conseqüência, há uma maior exigência quanto à transparência e a fidedignidade das informações prestadas por estes profissionais. Esta preocupação visa, primordialmente, manter a confiança de empresas, investidores, fornecedores e da sociedade, dentre outras, na responsabilidade ética do contador, denegrida pelo envolvimento nas fraudes detectadas. Desta forma, o presente estudo teve como objetivo verificar a conduta ética dos contadores e técnicos em contabilidade quando, no exercício de suas atividades profissionais, se depararem com questões relacionadas a fraudes. Para tal, são considerados alguns fatores que podem tanto vir a influenciar o processo decisório ético de um indivíduo, demonstrados através do Modelo de tomada de decisão, desenvolvido por Alves, quanto motivar um indivíduo a cometer um ato fraudulento, evidenciados através do Modelo denominado Triângulo da Fraude, desenvolvido por Cressey. Buscando responder a questão norteadora desta pesquisa, executou-se a análise descritiva e estatística dos dados, com a utilização de técnicas não-paramétricas. Para a análise descritiva dos dados foram elaboradas as tabelas de freqüências e calculadas as medidas de posição e dispersão, através do cálculo dos valores da média, moda, mediana e desvio padrão, quando aplicáveis. Em relação à análise estatística dos dados, foram utilizados os testes não-paramétricos de Spearman e a Regressão logística Multivariada. Os resultados demonstraram que a maioria dos profissionais de contabilidade, da amostra pesquisada, reconhece a questão moral inserida nos cenários, discorda dos atos dos agentes de cada cenário e, ainda, classifica esses atos como graves ou muito graves. Entretanto, verificou-se que esses profissionais de contabilidade tendem a ter um posicionamento mais voltado para a teoria teleológica, uma vez que a intenção de agir é influenciada por alguns fatores, como a oportunidade, a racionalização e, principalmente, a pressão. Alguns fatores individuais também apresentaram influência sob o posicionamento ético dos profissionais entrevistados nesta pesquisa. / Cases of fraud have occurred, often, in the world market. Several are involved in these cases, including the accounting class. The accounting scandals, especially the most famous, such as focusing on companies and Enron WordCom, kindled to greater concern about the ethical conduct of professional accounting. As a result, there is a greater demand on the transparency and reliability of information provided by these professionals. This concern is aimed, primarily, to maintain the confidence of businesses, investors, suppliers and society, among others, the ethical responsibility of the meter, denigrated by involvement in the fraud detected. Thus, this study aimed to verify the ethical conduct of accountants and accounting technicians in when, in the exercise of their professional activities, are confronted with issues related to fraud. This is considered some factors that can both come to influence the ethical decision making of an individual, demonstrated by the model of decision making, developed by Alves, as a motivated individual to commit a fraudulent act, as evidenced by the model called Triangle of fraud, developed by Cressey. Seeking to answer question guiding this study, performed to exploratory and confirmatory analysis of data, using non-parametric techniques. For exploratory data analysis were made tables of frequencies and calculated the measures of position and dispersion, by calculating the values of average, mode, median and standard deviation, where applicable. For the confirmatory analysis of data, were used non-parametric tests of Spearman and multivariate logistic regression. The results showed that the majority of accounting professionals, the sample, recognizing the moral issue included in the scenarios, disagrees the acts of agents of each scenario, and also classifies such acts as serious or very serious. However, we found that these accounting professionals tend to have a position more toward the teleological theory, since the intention to act is influenced by factors such as opportunity, rationalization and particularly the pressure. Some individual factors also had influence on the ethical position of the professionals interviewed in this research.

O ensino de Matemática é, de modo geral, orientado pelos processos contidos nos livros didáticos. Sendo assim, a organização dos conceitos matemáticos nesses livros deveria ser capaz de permitir ao leitor interpretar a Matemática em sua essência, admitindo o estabelecimento de relações entre os conteúdos. No entanto, o que geralmente se observa nos materiais é um aglomerado de definições e conceitos desconexos que conduzem o leitor a dificuldades de aprendizado na área. Por essa razão, a presente dissertação teve o objetivo principal de localizar, além de caracterizar, os pontos notáveis do triângulo: o centróide ou baricentro (G), o ortocentro (H), o circuncentro (O), o centro (N) da circunferência de nove pontos, os três ex-centros das circunferências ex-inscritas, as projeções ortogonais dos vértices sobre os lados opostos e os pontos de tangência da circunferência inscrita e ex-inscrita. Quatro abordagens são apresentadas em busca de tal objetivo: a-) apresentar a geometria do triângulo segundo técnicas de percepção visual; b-) caracterizar alguns pontos notáveis do triângulo, como pontos de máximo ou de mínimo de funções com as demonstrações utilizando desigualdade de Cauchy-Schwarz e entre média aritmética e geométrica; c-) utilizar um sistema cartesiano adequado para o cálculo das abscissas e ordenadas do centróide (G), do ortocentro (H) e do circuncentro (O) de um triângulo; d-) utilizar os números complexos para a completa localização de todos os pontos notáveis do triângulo além de apresentar a equação da reta de Euler, o incentro (I) e os três excentros IA, IB e IC localizados em fórmulas simples. A dissertação finaliza com o Teorema de Feuerbach, apresentado com uma prova elementar, mostrando que a circunferência de nove pontos e a circunferência inscrita são tangentes internamente e que a circunferência dos nove pontos é tangente exteriormente a cada uma das três ex circunferências e o Teorema de Napoleão, no qual os baricentros de triângulos equiláteros, construídos a partir dos lados de um triângulo qualquer, formam um outro triângulo equilátero. Comparando as várias abordagens da dissertação, a conclusão é a de que a compreensão dos números complexos paradoxalmente simplifica a resolução de problemas de geometria plana e a solução de equações polinomiais. Assim, acredita-se que uma maior exploração desse conteúdo no ensino da Matemática poderia tornar o aprendizado mais atraente e simplificado / The teaching of Mathematics is generally guided by the procedures contained in the textbooks. Thus, the organization of the mathematical concepts in these books should be able to allow the reader to interpret the Mathematics in its essence, admitting the establishment of relationships between the contents. However, what is observed in the materials is a conglomeration of disparate definitions and concepts that lead the reader to learning difficulties in the area. For this reason, this work aimed to locate and characterize the notable points of the triangle: the centroid or barycenter (G), the orthocenter (H), the circumcenter (O), the center (N) of circumference of nine points, three former centers of the ex-inscribed circles, orthogonal projections of the vertices on the opposite sides and the points of tangency of the inscribed and the ex-inscribed circumference. Four approaches are presented to achieve these goals: a-) to introduce the geometry of the triangle using visual perception techniques, b-) to characterize some notable points of the triangle, as points of maximum or minimum of functions with the demonstrations using the Cauchy-Schwarz inequality and between the arithmetic and geometric mean;-c) to use a suitable Cartesian system for calculating the abscissas and ordinates of the centroid (G), of orthocenter (H) and of the circumcenter (O) of a triangle;-d) to use complex numbers for the complete location of all notable points of the triangle, beyond depicting the Euler equation of the line, the incenter (I) and the three former centers IA, IB and IC located in simple formulas. The work is concluded with the Feuerbach\'s Theorem, presented with an elementary proof, showing that the nine-point circle and the incircle is tangent internally and that the circumference of the nine points is externally tangent to each of the three ex-inscribed circles and the Napoleons Theorem, in which the barycenters of equilateral triangles, constructed from the sides of any triangle, form another equilateral triangle. Comparing the approaches detached hitherto, the conclusion is that the understanding of complex numbers paradoxically simplifies troubleshooting of plane geometry and the solution of polynomial equations. Thus, it is believed that further exploration of this content in mathematics education could make learning more attractive and simplified

Although human panelists provide unparalleled data, they are prone to bias, which is a primary concern of sensory scientists. Motivational bias is of concern because it determines how much effort a panelist will exert to be consistent, find a difference, or use appropriate descriptors when taking a test. For central location tests, money has become a common motivational device to compensate panelists for their time and effort. Studies have documented that money can change results in sensory testing but have not measured the impact on motivation. Additionally, little research has been conducted on the effect of the test food itself as a motivator. This scientific investigation explored monetary reward and the test food type as motivational factors to examine how these affected untrained panelist effort. Panelist accuracy on a triangle test and assessment time to complete the triangle test were measured as the response variables. Two models were generated using the two response variables, and both were adjusted for panelist age, gender, liking, time of day, and day of the week. Statistical analysis indicated that monetary compensation was not a primary motivational factor for untrained sensory panelists, but might play a role in panelist attendance. Food type impacted models differently showing its importance but also making results inconclusive. Other factors like gender, age, time of day, and day of the week were significant to a panelist's motivation and may be related to things like food involvement and workday accumulation, but more research is needed to further support these ideas. In addition to the study at hand, three preliminary studies were completed prior to obtaining the results in the main study. First, surveys gathering data on general liking of different foods were conducted. Triangle tests were then completed on potential products to confirm that treatments created for foods were not too obvious for panelists. The last preliminary step was to confirm that all differences used for treatment differences per product type were similarly different, even though not similar in nature. Difference from control tests were conducted on each product and its treatments to find how different a product's treatments were from each other. Eventually, we were able to verify that between all treatment difference couples for all products were similarly different by keeping the individual differences between a range of 10 points or 10% of a 100-point scale.

Small businesses have an important role to play in the U.S. economy. However, employee fraud can jeopardize the sustainability of small businesses. Grounded on Cressey's fraud triangle theory, the purpose of this multiple case study was to explore strategies used by selected managers and owners of small retail businesses to detect and prevent employee fraud. Ten participants from 5 small retail businesses participated in the study. Nine participated in a face-to-face semistructured interview, and 1 participated in a telephone interview. These participants included 5 owners and 5 managers of small retail businesses in the state of Michigan in the United States who have implemented strategies to detect and prevent employee fraud. Through a process of methodological triangulation, casual observations and documentary evidence supplemented data collected through semistructured interviews. Using thematic analysis by coding narrative segments, the research findings included themes of controls and communication, cash register accountability, segregation of duties, monitoring, and action against perpetrators. Managers and owners of small businesses may benefit from the findings of this study by gaining awareness of the need to detect and prevent employee fraud. The implications for positive social change may include the potential to increase appropriate controls over employee fraud, thus enabling owners of small retail business an opportunity to operate effectively and efficiently, which could increase employment opportunities. Increased employment opportunities could create a positive effect on other small retail businesses and allow local communities to prosper.

A lot has been said about the Fibonacci Convolution Triangle, but not much has been said about the Tribonacci Convolution Triangle. There are a few ways to generate the Fibonacci Convolution Triangle. Proven through generating functions, Koshy has discovered the Fibonacci Convolution Triangle in Pascal's Triangle, Pell numbers, and even Tribonacci numbers. The goal of this project is to find inspiration in the Fibonacci Convolution Triangle to prove patterns that we observe in the Tribonacci Convolution Triangle. We start this by bringing in all the information that will be useful in constructing and solving these convolution triangles and find a way to prove them in an easy way.