Question

Theorem 10.1 and theorem10.2

Answer 1

10.1 theorem

important theorem

completely detailed.......

Answer 2

                        $\huge{\underline{\mathfrak{Theorem\::\:10.1\:&\:10.2}}}$

Theorem 10. 1 states that Equal chords of the circle subtend equal angles at the center.

                       Have a look in the attachment for the diagram.

• Given : AB = CD ( These are the equal chords )

• To Prove : ∠AOB = ∠COD.

• Proof :

                   

                      In Δ AOB & Δ COD,

                AO = CO ( radius of the circle ) --- Side

               AB = CD  ( already given ) --- Side

               BO = DO ( radius of the circle ) --- Side

Using Side Side Side congruence ( SSS ), we can say that Δ AOB ≅ Δ COD

Using CPCT ( Congruent parts of corresponding triangles ), ∠AOB = ∠COD.

                   ∴ The given angles are proved to be equal.

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Theorem 10.2 states that If the angles subtended by the chords of a circle are equal, then the chords are equal.

Have a look at the diagram in the attachment.

• Given : ∠AOB = ∠COD.

• To Prove : AB = CD.

• Proof :

                     In Δ AOB & Δ COD,

                AO = CO ( radius of the circle ) --- Side

             ∠AOB = ∠COD ( already given ) --- Angle

               BO = DO ( radius of the circle ) --- Side

Using Side Angle Side congruence ( SAS ), we can say that Δ AOB ≅ Δ COD

Using CPCT ( Congruent parts of corresponding triangles ), AB = CD.

                   ∴ The chords of the circle are proved to be equal.