**Introduction**

Geometry is a one of the part of mathematics. Mathematical Geometric statement considered the sequence of statements by proofs, every statement being adjusted with some definition or a proposition or an axiom that is before launched by the method of deduction using only the permitted logical rules. In Geometric proof, when the condition should be solved analytically to show the given condition should be always true, the condition must be true under any circumstances. There are some predefined theorems to show the given condition as a true statement.

**Geometry proof 1**. The angles opposite to equal sides of a triangle are equal

**Given: **ABC is a triangle where AB = AC

**To prove: ∠B = ∠C.**

**Construction:** Mark the mid point of BC as M and join AM.

**Proof:** In the triangles AMB and AMC

(i) BM = CM (ii) AB = AC (iii) AM is common.

By the SSS criterion, ΔAMB ≡ ΔAMC.

so, The angles are equal. In particular, ∠B = ∠C.

Hence the theorem is proved.

**Proof 2:**

**Total Vertical angle of circle is 360 degree and Each angle should be equal**

Here total angle of circle is 360 degree

Now we have to proof total vertical of circle is 360 Degree that mean A=90 Degree B= 90 Degree, C= 90 Degree, D=Degree

In given diagram A+B+C+D=360 Degree

In geometry theorem Total angle of circle is 360 degree

In circle four quad parts and angles are equal

So now we have to write A=B=C=D

B=C=D ->A (A=B, B=A, C=A,D=A)

A+B+C+D=A+A+A+A

360=4A

4A=360 Degree

So A=360/4

A=90 Degree

A=90 Degree=90Degree, C=90Degree, D=90 Degree (Because A=B=C=D)

Hence proof.

**Proof 3: If the two straight lines are parallel, then their slopes are equal.**

Since the two straight lines are parallel, θ = 0. ∴ tan θ = 0

⇒ m_{1} − m_{2} / (1 + m_{1}m_{2}) = 0

⇒ m_{1} − m_{2} = 0

i.e. m_{1} = m_{2}

the straight lines are parallel, then the slopes are equal.

**Example in geometry problem**

Find the distance between the points A(−15, −3) and B (7, 1).

**Sol :** Let d be the distance between A and B.

Then d (A, B) =√[( x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}]

= √[(7+15)^{2} + (1+3)^{2}]

= √[(22^{2} + 4^{2})]

= √[(484 + 16)]

= √500

= 10√5