# i want all the solution of these theorems aas soon as possible1.prove that sum of three angles of a triangle is 180 degree2.Angle opposite to equal sides of a triangle are equal3.Two triangles are congurent,if any two angles and included side of  one triangle is equal to any two angles and the included side of the other triangle.4.If two lines intersect at each other at point o.Prove that they are vertically opposite angles.

Hi!
Assuming you want the answer for the theorem;

Prove that sum of three angles of a triangle is 180 degree

Here is the answer to your query.

Consider a triangle PQR and ∠1, ∠2 and ∠3 are the angles of Δ PQR (figure shown below). We need to prove that ∠1 + ∠2 + ∠3 = 180°. The straight lines AB and CD intersect in O.
∴ ∠AOC + ∠AOD = 180°   … (1)  (Linear pair)
∠AOD + ∠BOD = 180°    … (2)  (Linear pair)
From (1) and (2), we get
∠AOC + ∠AOD = ∠AOD + ∠BOD
∠AOC = ∠BOD
Similarly, ∠BOC = ∠AOD

Thus, if two straight intersect each other, then the pair of vertically opposite angles are equal.

Cheers!

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1. TO PROVE THAT SUM OF 3 ANGLES OF A TRIANGLE IS 180 DEGREE

DRAW A TRIANGLE ABC . MAKE A LINE PARALLEL TO THE BASE . Since lines a and b are parallel, <)BAC = <)B'CA and <)ABC = <)BCA'.

It is obvious that <)B'CA + <)ACB + <)BCA' = 180 degrees.

Thus <)ABC + <)BCA + <)CAB = 180 degrees. R

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