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Yes, the statement and it's opposite are both true. This holds for every geometric fact.

Actually the converses are derived from statements only.

• Statements : These are the well defined proved laws in mathematics which gives an idea about something in the written form.

• Converses : These are the derived expressions from the originating statements just by shifting the position of the ideal words in that statement.

Usually it is said that is A → B is the statement, then it's converse will be B → A. All the above given statements are true with their converses.

Now let's understand to the statements and converses above.

Statement - In a cyclic quadrilateral, sum of two opposite angles is always equal to 180°.

Converse - If the sum of two opposite angles of a quadrilateral is 180°, then the quadrilateral is a cyclic quadrilateral.

If we see the relationship in both the statement and converse, we get that only there has beeb shifting of some words. Intially they point out to something proved and same but the way in which they are expressed varies.

This can help us in proving other things also.

Statement - Pythagoras Theorem states that in a right angled Triangle, the square of Hypotenuse is equal to the sum of squares of other two sides.

Converse - If the sum of squares of other two sides of the triangle is equal to the hypotenuse, then the triangle is right angled triangle.

Here we see that both the statement and the converse statement that a right angled satisfies the condition of Pythagoras Theorem that is square of Hypotenuse is equal to the sum of squares of other two sides.

• Other Geographical Facts ::

Yes, this condition is true for other geographical facts too. The reason because both converse and statement prove a single thing in different ways and they are proved on themselves.

• 'If and only if' condition ::

'If and only if' condition is applied just to show a definite relationship. Actually in order to prove a statement or converse, we need certain conditions. These conditions are determined using 'if and only if' conditions.

For example, Pythagoras Theorem is proved if and only if sum of other two sides is greater than the hypotenuse of a triangle.

Sum of opposite angles of a cyclic quadrilateral is 180°

Let ABCD is cyclic quadrilateral.

To prove that ∠A+∠C=180° and ∠B+∠D=180°

Construction: Join OB and OD.

We know that ∠BOD=2∠BAD ...(The angle at the centre is twice the angle at the circumference )

⟹ ∠BAD= 21

∠BOD.

~Similarly, we can write ∠BCD= 21

∠BOD.

⟹Now, add above two equations,

⟹∠BAD+∠BCD= 21

⟹ ∠BOD+ 21

⟹ ∠DOB

⟹ ∠BAD+∠BCD= 21

(∠BOD+∠dob) ...(complete angle)

⟹∠BAD+∠BCD= 21 ×360°=180°.

So we get, ∠A+∠C=180°

And similarly ∠B+∠D=180°.

Hence,option C is correct.

Introductory plane geometry involving points and lines, parallel lines and transversals, angle sums of triangles and quadrilaterals, and general angle-chasing.

Introductory plane geometry involving points and lines, parallel lines and transversals, angle sums of triangles and quadrilaterals, and general angle-chasing.Experience with a logical argument in geometry written as a sequence of steps, each justified by a reason.

The converse of Pythagoras theorem states that “If the square of a side is equal to the sum of the square of the other two sides, then triangle must be right angle triangle”. Whereas Pythagorean theorem states that the sum of the square of two sides (legs) is equal to the square of the hypotenuse of a right-angle triangle. But, in the reverse of the Pythagorean theorem, it is said that if this relation satisfies, then triangle must be right angle triangle. So, if the sides of a triangle have length, a, b and c and satisfy given condition a2 + b2 = c2, then the triangle is a right-angle triangle.