Geometry doesn’t have to be complicated and filled with mind-breaking theorems. **Mathematics is actually quite simple** when you discover the keys that unlock its logical paths. Some things never change and that is also the case in **geometry**.

We have compiled a list of the most common geometrical subjects and the simple way to solve them. These tricks will actually help you understand geometry better and offer you a more clear perspective on problems.

## About Angles…

- If the sum of any two angles is 90°, then the two angles are
**complementary angles****.** - If the sum of any two angles is 180°, then the two angles are
**supplementary angles****.** - The sum of all interior angles of a triangle is 180° and the sum of all exterior angles of a triangle is 360°.
- In a parallelogram, two opposite angles are equal.

## About Triangles

– The sum of any two sides of a triangle is greater than the third side. And the difference between any two sides of a **triangle** is less than the third side.

– The orthocenter, the centroid, and the circumcenter of a triangle are all on a straight line, which is called the **Euler line****.**

– The **Triangle of Pascal** is a triangular array of numbers where each number on the “interior” of the triangle is the sum of the two numbers directly above.** **

** **

– A diagonal line cuts a parallelogram in two congruent triangles.

## About Squares

- A square has 4 sides of equal length, thus it is also called a regular
**quadrilateral**. - However, amongst all the other quadrilaterals with the same perimeter, the square has the largest area.
- Different from triangles, the sum of the internal angles of a square is 360°.
- A cube is a 3D square.
- A square is also a rhombus with right angles and a rectangle with equal sides.
- The perimeter of a square is 4 times the length of one side.

## About Circles

- Any point on the edge of a circle is at an equal distance from the center of the circle;
- Circle actually refers to the boundary of the shape while the disk is used to refer to the whole shape, including the inside.
- A circle has the shortest perimeter of all shapes with the same area.
- The unit circle is centered at the coordinates (0,0) and can be used to model any
**trigonometric function**. - When two lines intersect inside a circle, then the measures of the segments of each line multiplied with each other is equal to the product from the other line.

## Common Knowledge About Geometry

Geometry is a very old, almost ancient discipline. It was used to build up the pyramids, entire infrastructures of lost civilizations and it’s a part of our daily lives as everything is built on either circles or squares. Hence, it is very important and useful to study geometry, at least at a basic level. In order to understand how fascinating geometry is, take a look at this historical facts:

- The compass is one of the oldest and most used tools in geometry;
- Babylonians discovered the measurement of a circle which was approximately 3 times the diameter. It is very similar to the measurement of Pi (around 3.14).
- Leonardo Da Vinci and Albrecht Dürer have discovered ways to represent 3D objects on 23 plans. Thus, from the 19-20th centuries further research has been done on the subject leading to projective geometry used nowadays in computer graphics.
- The
**measurement**of paper sizes (e.g. A4) use a 1:√2 ratio. If you cut the sizes in half crosswise, the same ratio will be maintained. It’s a great tool for scaling. - Banach–Tarski paradox states that if a 3D object is decomposed into smaller pieces, those pieces when put together can create two identical copies of the original object. Do you think it’s true?
**Eratosthenes**estimated the Earth’s circumference using basic mathematics and he made it with a high precision.- Apparently, followers of the Pythagora school used small rocks to represent numbers while solving mathematical equations. Calculus means pebbles in Greek. Can you try to solve a mathematical problem using
**rocks**or pebbles? - Geometry means “geo” – earth and “metria” – measurement, from Greek.

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