# Geometry in Our World

By incorporating multiple theorems explored in class, we hope to thoroughly display our awareness and knowledge surrounding this diverse topic.

## Geometric Properties of Glass Building

The glass building we have chosen exemplifies the theorems we have used this geometry unit. Not only does this image show many of the theorems we have previously learned, but it also shows many geometric shapes, such as a rhombus.

## Sum of Interior Angles

I know that a quadrilateral is a shape that is based off of four straight sides and angles. The picture we have chosen demonstrates the geometric shapes we have been learning about. Knowing that a quadrilateral angles sum equal 360° , while calculating this must be the sum.

1. <CDE= 130.87°

<DEF= 49.13°

<EFC= 130.87°

<FCD= 49.13°

These angle measurements all add up to 360 degrees which is the correct amount for a quadrilateral. This profoundly displays the sum of the interior angles theorem.

130.87° X 2 = 261.74°

49.13° X 2 = 98.26°

261.74°

+98.26°

=360

## Corresponding Angles Theorem

Corresponding angles are formed when a transversal passes through two lines. The angles that are in the same position in terms of the transversal are called corresponding angles.

1. We believe that the angles are connected from the transversal. This told us that <CDE= 130.87° was equal to <EFC which was also 130.87°
2. The same rule applied to the other corresponding angles, which were <DEF= 49.13°. This angle was opposite (also known as corresponding) was <FCD= 49.13°

3) Supplementary Angles Theorems

Two angles are supplementary angles when they both add up to 180°. This does not matter whether they are next to each other or not. Angles EFC and DEF are supplementary to each other, while angles CDE and FCD are supplementary to each other.

When we added 130.87° + 49.13°, it equaled 180° , therefore this theorem is applicable

Examples:

60° and 120° are supplementary angles.
93° and 87° are supplementary angles.

## Geometric Properties of An Hourglass

This hourglass is symmetrical, therefore all measurements are mimicked at the bottom as they are at the top. Consisting of 2 isosceles triangles, there should only be 2 different measurements, although there are 6 angles to be measured.

Angle BCA= 53.13°

Angle DCE= 53.13°

*OPPOSITE ANGLE THEOREM*

Angle CED= 180° - 53.13°

=126.87° ÷ 2

=63.43°

*SUM OF INTERIOR ANGLES THEOREM*

Angle EDC= 63.43°

Angle CAB= 63.43°

Angle ABC= 63.43°

*OPPOSITE ANGLE, ISOSCELES TRIANGLE, AND SYMMETRICAL THEOREM*

Undoubtedly, this hourglass contains many sets of parallel lines. Within these parallel lines, we discover patterns that help us determine the measures of each angle, such as the "Z" pattern. This picture also includes corresponding angles, which occupy the same relative position at each intersection where a straight line crosses two others. If the two lines are parallel, the corresponding angles are equal. The mage below clearly illustrates the "Z" pattern. This proves that opposite angles that intersect are equal, (since angles CED and EDC both equal 63.43°) the "Z" pattern has been applied, due to the formation the hourglass lies in.

## Geometric Properties Within Construction

With the ladder leaning against the house, it's clear that a scalene triangle is created. The Pythagorean Theorem is applied because if we did not know the length of a certain side, we could easily use or knowledge to figure it out.

*a2 + b2 = c2

By knowing this, we can make sure all measurements are accurate.

a2 + b2 = c2

=15.5 cm2 + 10.5 cm2 = 18.72 cm2

=240.25cm + 110.25cm = 350.5cm

=350.5cm = 350.5cm

## Geometric Properties of a Trampoline

In this regular polygon, all angles are equal. Regular polygons have 6 triangles within them. In order to determine the sum of the interior angles, we can simply add up every angles within the shape, or multiply 180° By 6, Since each triangle has a sum of 180°. Thankfully, in class we learned a much quicker and efficient method of determining the sum of any regular polygon.

*The formula is 180°(n-2)

The "N" represent the amount of sides. We must subtract 2 from whatever this number may be, because this will give you the amount of triangles in that particular shape.

180°(n-2)

=180°(6)

=1080°

In order to determine the measure of a single angle, we can divide this sum by the total amount of sides in the shape.

1080°÷8= 135°

Therefore, each interior angle in a regular octagon equals 135°

## Mathematical Analysis of Supplementary Angles

The blue line (QN) represents supplementary angles. Supplementary angles always equal 180° when added together, and are formed on a straight line. Since the sum of these 4 angles equal 180° , they are undoubtedly supplementary to one another.

## Mathematical Analysis of Isosceles Triangle Theorem

The isosceles triangle is made up of two adjacent lines that connect at the top of the triangle to create an angle, and both lines are the same length. Both angles equal 38.02°because since both side lengths are the same and connect at a common vertex, both angles are the same. The 2 lines on each line show you that both angles and lines are the same. This pyramid is an isosceles triangles because 2 out of the 3 sides are equal, and when added together, all angles in the pyramid equal 180° . (103.97° + 38.02° + 38.02° = 180° )

## Geometric Properties of a Bedframe

Using our creativity, we both have spotted 2 angles that come together to form the perfect "Z" pattern, due to it's parallel lines. This picture includes corresponding angles as well.

Corresponding Angles - the angles that occupy the same relative position at each intersection where a straight line crosses two others. If the two lines are parallel, the corresponding angles are equal.

View the image below for more measurements and angles. This proves that opposite angles that intersect are equal, (since both angles equal 42.89°) the "Z" pattern has been applied, due to the formation the metal frame lies in.

## Geometric Properties in Tiles

Parallel lines are everywhere, and it's quite clear to pick them out in this picture. This tile pattern forms the "C" pattern, where the top and bottom angles in the pattern equal the same amount. As shown, both angles FDC and DCE are 90°, also known as right angles.

## In Conclusion...

Overall, we hope that by thoroughly exploring and exemplifying our knowledge, you can grasp the importance of geometry, and how exposed we are to it in our everyday lives. Through multiple theorems, it is quite clear that having this wide-ranging intelligence in this topic is essential in order to live a life of awareness and appreciation for mathematics.