Now we will discuss some common Euclid axioms or notions. Axiom means statements that do not require proof. These Euclid axioms are not restricted to geometry. They are valid for basic arithmetic operations including addition and subtraction.

6 Euclid Axioms

1. The first Euclid axiom states that things which are equal to the same thing are equal to one another.

For example, if an area of a triangle equals the area of a rectangle and the area of the rectangle equals that of a square, then the area of the triangle also equals the area of the square.

This means an area of a rectangle is equal to the area of the triangle and the area of a triangle is equal to the area of a square, which implies that the area of a rectangle is equal to the area of triangle is equal to area of square.

2. The second axiom states that if equals are added to equals, the wholes are equal.

For example, we have two lines AB and CD, which are equal.

If we add a line segment MN to both of the lines, the resulting lines AB plus MN and CD plus MN are equal in length since the lines AB and CD are equal, which states the axiom perfectly.

3. Third Euclid is axiom almost the same as the second axiom. This axiom states that if equals are subtracted from equals, the remainders are equal.

For example, if we subtract line segment MN from two equal lines AB and CD, the resulted lines AB minus MN and CD minus MN are also equal.

4. The Fourth Euclid axiom states that things which coincide with one another are equal to one another.

For example, two congruent triangles ABC and XYZ coincide with one another, this means their corresponding sides and angles are equal.

5. Now Euclid axiom 5 states that the whole is greater than the part.

This square ABCD is made up of two triangles ABD and BCD.

Area of square ABCD is the sum of areas of triangles ABD and BCD.

Symbolically we can write that the area of square ABCD is greater than the area of triangle ABD, which means that there is area of triangle BCD such that area of square ABCD is the sum of areas of triangle ABD and BCD.

6. The sixth Euclid axiom states that things which are double of the same things are equal to one another.

For example, we have given 2 lines AB and CD, which are equal.

If we double these lines then 2AB and 2CD are also equal. And the last and seventh axiom states that things which are halves of the same things are equal to one another. For example, we have given 2 lines AB and CD, which are equal. If we half these lines then half of AB and half of CD are also equal.

What are Euclid's axioms used for?

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