# Geometry in our world

## HERE IS THE PICTURE WE DECIDED TO CHOOSE FOR OUR TASK.

With help of a program GeoGebra and our creativity, we have created 7 geometric properties also known as theorems. We hope you enjoy!

## sum of the interior angles of a trapizoid

Sum of interior Angles of a Trapezoid Theorem

Sum=180° (Number of Triangles)

=180° (n-2)

=180°(4-2)

=180°(2)

=360°

Application of Theorem on Trapezoid

m<ABC= 112.75°

m<BCD=103.08°

m<CDA=43.61°

m<DAB=100.56°

(when you add up these interior angles you will get 360)

Therefore, any missing angle in the trapezoid can be found using the sum of interior angles of polygon (Trapezoid in this case) Theorem.

## sum of the interior angles of a triangle

Sum of interior Angles of a Trapezoid Theorem

Sum=180° (Number of Triangles)

=180° (n-2)

=180°(4-2)

=180°(2)

=360°

Application of Theorem on Trapezoid

m<ABC= 112.75°

m<BCD=103.08°

m<CDA=43.61°

m<DAB=100.56°

(when you add up these interior angles you will get 360)

Therefore, any missing angle in the trapezoid can be found using the sum of interior angles of polygon (Trapezoid in this case) Theorem.

## Parralel line therom shown through the Z pattern

Parallel line theorem

Line segment AF is parallel to Line segment CB.

The angles make a "Z" pattern with the transversal line segment, hence the name.

m<FAB=25.09°

m<ABC=25.08°

Therefore, to find a missing angle in the parts shown in the picture above, using the Parallel Line Theorem- Alternate Angles (Z Pattern) would be the right option.

## Supplementary angle theorem

Supplementary angle theorem

BGHF is a straight angle.

The line splitting it is creating two separate angles which both have a sum of 180° while the straight angle stays the way it was.

m<AHF=105.94°

m<FHB=74.06°

105.94+74.06= 180°

Therefore, to find a missing angle in the parts shown in the picture above, use the supplementary angle theorem. (It may look like the parallel line theorem but these line segments helped us make the lines).

## Opposite angle theorem

Opposite angle theorem

Line segment AB intersects Line segment EF at point H.

Opposite angles have the same measures. They are usually across from each other.

m<AHF=105.94°

m<EHB=105.94°

m<AHF and m<EHB are the same and across from each other.

Therefore, the angles in the picture above can be found using the opposite angle theorem.

## Exterior angle theorem

Exterior Angle Theorem

Add the angles adjacent to the exterior angle to find the missing angle.

<CAB+ <ABC= <ACD

<CAB= 7.47°

<ABC= 101.51°

7.47°+101.51°= 108.98°

If the exterior angle is given and you have to find <ACB, you would need to use the supplementary angle theorem but the exterior angle would come to great use in this situation.

Therefore, you would need to use the exterior angle to find the missing angle (exterior angle).

## Complementary angle theorem

Complementary Angle Theorem

The sum of a complementary angle should be 90° (Right angle).

1. <ABD+<DBC= 90° (Right/ complementary angle)

Ø 45+45= 90°

2. 90°(Right/ complementary angle) - <ABD =<DBC

Ø 90 - 45=45°

3. 90°(Right/ complementary angle) - <DBC= <ABD

Ø 90 - 45=45°

Therefore, to find a missing angle in the picture above, you would have the use the complementary angle theorem.