Geometry in our world

By: William and Avkash

Hearst Tower (Manhattan)

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The building that we chose to do the "Geometry in our World" task on is the Hearst Tower in Manhattan. Through the use of Geogebra and Smore, we will demonstrate nine of its geometric properties. Following is the simplified version of the building face that we have used.

Simplified:

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Supplementary Angles

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One example of a supplementary angle in this picture is, <ACD and <ACE. This theorem states that two angles that add up to 180° are called supplementary angles. In this picture <ACD is 68° and <ACE is 112°. The sum of these two angles is 180°, causing them to be an instance of supplementary angles.

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Opposite Angle Theorem

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One example of opposite angles are <ACD and <ECF. The theorem states that angles on opposite sides of two intersecting line segments must have the same measurement. As <ACD and <ECF are on opposite sides of the intersection C, made from line segments DE and AF, they should be congruent. As both these angles are 68° we can prove that this theorem is true.

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Sum of the Interior Angles - Triangle

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An example of this is the triangle ACD. The theorem states that the sum of the angles in any triangle must equal to 180°. In this picture when you add up each of the angles of said triangle: <ACD which is 68°, <ADC which is also 68°, and <DAC is 45°, the sum is 180°.

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Sum of Interior Angles Formula - Hexagon

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One example of the sum of the interior angles formula being used is through the hexagon EHIJKL. The formula stated above is 180°(n-2) with n being the amount of sides in the shape. As shown in the image above, if we follow through with this formula using a hexagon which has 6 sides, then the sum of the interior angles of a hexagon must be 720°. Shown beside that is the sum of the angles of hexagon EHIJKL which is also 720°. As both of these are the same, we can prove that this theorem is true.

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Isosceles Triangle Theorem

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An example of this is the isosceles triangle ACD. This theorem states the angles opposite of the equal sides in an isosceles triangle must be congruent. In this example of triangle ACD, <ACD and <ADC are both 68° as well as being opposite to the equal sides line segments AC and AD.

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Exterior Angle Theorem

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One example of the exterior angle theorem is using the triangle ACD with <ACE being the exterior angle. This theorem states that sum of the two non-adjacent angles inside the triangle are equal to the exterior angle. In this case, the non-adjacent angles are <CAD and <ADC which are 45° and 68° respectively. When added up, these equal 112° which is also the measurement of <ACE. Through this we can prove that this theorem is true.

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Parallel Line Theorem – Corresponding Angles (F Pattern)

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One example of corresponding angles are <BAC and <ECF. When highlighted, these two angles form the shape of an F. This theorem states that corresponding angles, the angles created at the same location at each intersection when a transversal crosses parallel lines, must be congruent. In the example above, line AF is the transveral with lines AD and CE being the parallel line segments that it crosses through. <BAC and <ECF are both the bottom-right angle of their respective intersection and both are 68°. This enforces the theorem.

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Parallel Line Theorem – Alternate Angles (Z Pattern)

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One example of alternate angles are the <DCG and <CGF. This theorem states that alternate interior angles, ones on opposite sides of opposite ends of the transversal between two parallel lines, which also happen to form the shape of a Z, must be congruent. This is displayed when parallel line segments: DE and GF are split by transversal CG, creating <DCG and <CGF which are both 67°. This proves the theorem.

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Parallel Line Theorem – Co-Interior Angles (C Pattern)

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An example of co-interior angles would be <ECF and <CFL which make the shape of a C hence the name C pattern. This theorem states that co interior angles, which are created in between the parallel lines and on the same side of the transveral, must equal to 180°. In this image EC and FL are a set of parallel lines but are cut by the line segment CF and when the angles are added (68° and 112°) it does equal 180°.