2.2 角度和三角
Section outline
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House Design
::房屋设计Engineering Question : The image below is a basic outline of a house. Engineers face the problem of needing to put in support beams without taking up unnecessary space in the rooms . How can you make the roof in the image stronger without adding supports that use up space inside the rectangular living area of the house?
::工程问题:下面的图像是房子的基本轮廓。 工程师们面对的问题是如何在不占用室内不必要的空间的情况下安装支撑梁。 您如何使图像中的屋顶更坚固,而不增加在房屋矩形居住区内占用空间的支撑?INTERACTIVERoof DesignTry to make the roof more stable by creating line segments to support it!
::试着通过创建支持屋顶的线段来让屋顶更加稳定 !+Do you want to reset the PLIX?Triangles are the strongest shape because of their ability to redistribute pressure equally throughout the shape. This property of triangles is why they are so commonly used in engineering and architecture and why most houses have a triangular roof. However, even triangles have their limits. To keep triangular roofs from collapsing under heavy snow or extreme weather, engineers use even more triangles in the form of roof trusses. A roof truss is a framework made of triangles used to provide support to a triangular roof to prevent it from caving in.
::三角形是最坚固的形状, 因为它们有能力在整个形状中平均再分配压力。 三角形的这种属性就是为什么在工程和建筑中经常使用三角形, 以及为什么大多数房屋都有一个三角形的屋顶。 但是, 即使是三角形也有其极限。 为了防止三角形的屋顶在大雪或极端天气下崩溃, 工程师们使用更多的三角形, 即屋顶的三角形。 屋顶的三角形是一个三角形框架, 用来为三角形的屋顶提供支持, 以防止三角形的屋顶被切入 。Roof trusses made of triangles In this lesson, you'll explore how properties of triangles are used in the construction of houses.
::在这个教训中,你会探索 三角形的属性是如何 用于建造房屋的。
The Triangle Sum Theorem
::三角苏门神话There are several properties that triangles have in common: all triangles have three sides, all triangles have three angles , and the sum of all the angles in a triangle is always the same.
::三角形有几个共同的属性:所有三角形都有三边,所有三角形都有三个角度,三角形中所有角度的总和总是一样。Use the interactive below to learn or review the proof of the sum of all the angles in a triangle.
::使用下面的交互效果来学习或审查三角形中所有角度之和的证明。INTERACTIVETriangle Sum Theorem-
Move the red points to change the positions of the vertices.
::移动红点以改变顶点的位置。
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The Crown Molding Problem
::王冠的死期问题Crown molding is a wooden board that runs along the top of the ceiling at an angle. There is normally a decorative pattern carved into the wood. It can often have a frieze pattern carved into it. Carpenters need to be able to determine the angle at which to cut both sides of the board so that it fits in perfectly. The angle that runs along the wall is called the spring angle and the angle at which the ceiling meets the wall, often 90°, is called the wall angle.
::皇冠模具是一种木板,从一个角度沿着天花板的顶部运行,通常在木头上刻有装饰性图案,通常可以刻有碎屑图案。木匠需要能够确定切开板两侧的角,使其完全合身。在墙上运行的角称为弹簧角,天花顶与墙接合的角,通常为90°,称为墙角。Crown molding Carpenters need to know the Triangle Sum Theorem to calculate the angle at which to cut the board for the molding. The Triangle Sum Theorem states that the sum of the interior angles of a triangle is always 180°.
::Carpenters 需要了解三角形 Sum 理论, 以计算切开模型板的角度。 三角形 Sum 理论指出, 三角形的内部角总和总是 180 ° 。Example
::示例示例示例示例The crown molding in the picture is to sit against a 90° wall with a spring angle of 38°. At what angle does the other side of the molding meet the wall?
::图片中的皇冠模样是坐在90度的墙上,弹簧角为38度。Since the sum of the interior angles of a triangle is 180°, you can use the equation
::由于三角形的内角和内角为180°,您可以使用方程38+90+x=180。
::38+90+x=180128+x=180x=52Use the interactive below to find the missing angle in the triangles.
::在三角形中找到缺失的角度 。INTERACTIVEFind the Missing Angle-
Enter the missing angle into the input box
::在输入框中输入缺失的角度 -
Press next to move to the next problem
::按在下一个问题下移到下个问题的旁边
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Exterior Angles
::外外角An exterior angle is an angle formed by one side of a polygon and the extension of its adjacent side. To get a better idea of what this definition means, take a look at the picture below.
::外部角度是多边形的一面及其相邻侧延伸形成的一个角度。 要更好地了解这个定义的含义, 请看看下面的图景 。The outside angles of a triangle Use the interactive below to explore the exterior angles of triangles.
::使用下面的交互作用来探索三角形的外部角度。INTERACTIVEExterior Angle Sum Theorem-
Drag the red points to change the triangle.
::拖曳红色点以更改三角形。 -
Drag the slider and make observations about the exterior angles of a triangle.
::拖曳滑块,并观察三角形的外部角。
+Do you want to reset the PLIX?Discussion Question
::讨论问题What do you notice about the exterior angles of all triangles?
::你注意到所有三角形的外角吗?When designing a roof truss, you will need to use exterior angles in addition to similar math used in the crown molding example. Notice that e xterior angles are always a supplement to their adjacent interior angle, meaning that if you know one angle, you can find the other by subtracting it from 180.
::在设计屋顶三角形时,除了冠状模型中使用的类似数学外,还需要使用外部角度。请注意,外部角度总是其相邻内角的补充,这意味着,如果您知道一个角度,您可以从180个角度中减去另一个角度。In the interactive below, u se your knowledge of exterior angles and the Triangle Sum Theorem to identify all the missing angles in the roof truss.
::在下文互动部分,利用你对外部角度和三角SumTheorem的认知,查明屋顶三角形中所有缺失的角。INTERACTIVERoof Trusses-
Select the correct angle measurements from the drop down menus.
::从下拉菜单中选择正确的角度测量。 -
Check your answers with the Check button.
::使用“ 检查” 按钮检查您的答案 。 -
Note that the truss is vertically symmetrical.
::请注意,三角形垂直对称。
+Do you want to reset the PLIX?Discussion Question
::讨论问题Prove that the exterior angles of a triangle always sum to 360 degrees.
::证明三角形的外部角总等于360度。Summary
::摘要-
According to the Triangle Sum Theorem, the sum of the interior angles of a triangle will always equal 180°.
::根据三角理论,三角形的内部角总和总是等于180度。 -
Exterior angles are formed by one side of a polygon and the extension of an adjacent side
.
::外角由多边形的一面和相邻侧的延伸形成。 -
The sum of the exterior angles of a triangle will always equal 360
°.
::三角形外角的总和将永远等于360°。 -
An exterior angle and its adjacent interior angle
form a straight line
, so they are supplementary
.
::外部角及其相邻的内部角形成一条直线,因此是补充性的。
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Move the red points to change the positions of the vertices.
