Cyclic Quadrilateral (and Properties)

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Cyclic Quadrilateral (and Properties) A quadrilateral is called a cyclic quadrilateral if all four of its vertices lie perfectly on the circumference of a single circle. In other words, the points A, B, C & D are concyclic points. We have some properties related to Cyclic Quadrilaterals – which are also theorems Theorem 11: The sum of two opposite angles of a cyclic quadrilateral is 180°. We can also say that In a cyclic quadrilateral, opposite angles are supplementary Here, ∠ A + ∠ C = 100° + 80° = 180° ∠ B + ∠ D = 72° + 108° = 180° Let’s look at the proof Proof is here Theorem 11 - Chapter 5 Class 9 – Ganita Manjari Part 1 The converse of Theorem 11 is our next Theorem Theorem 12: If two opposite angles of a quadrilateral add up to 180°, then the vertices of the quadrilateral lie on a circle, i.e., they are concyclic. We can also say that If the sum of a pair of opposite angles of a quadrilateral is 〖𝟏𝟖𝟎〗^∘, the quadrilateral is cyclic. This gives us a mathematical test to prove if 4 scattered points lie on a single circle. Thus, to check if A, B, C, D are conclyclic, we check Does ∠𝑨+∠𝑪=〖𝟏𝟖𝟎〗^∘ ? Does ∠𝐁+∠𝐃=〖𝟏𝟖𝟎〗^∘ ? If yes, then points are concylic, and they form a cyclic quadrilateral Let’s look at the proof Proof is here Theorem 12 - Chapter 5 Class 9 – Ganita Manjari Part 1