In the given figure, AB is a chord of the circle with centre O and BT is a tangent to the circle. If ∠OAB = 35°, find the values of x and y.

In △OAB,

OA = OB (∵ both are radius of the common circle.)

So, △OAB is a isosceles triangle with,

∠OBA = ∠OAB = 35°.

Since sum of angles in a triangle = 180°.

In △OAB,

⇒ ∠OBA + ∠OAB + ∠AOB = 180°

⇒ 35° + 35° + ∠AOB = 180°

⇒ 70° + ∠AOB = 180°

⇒ ∠AOB = 180° - 70°

⇒ ∠AOB = 110°.

Arc AB subtends ∠AOB at centre and ∠ACB at remaining part of circle.

∴ ∠AOB = 2∠ACB (∵ angle subtended at centre is double the angle subtended at remaining part of the circle.)

⇒ 110° = 2y

⇒ y = $\dfrac{110}{2}$

⇒ y = 55°.

From figure,

∠ABT = ∠ACB = 55° (∵ Angles in alternate segments are equal.)

∴ x = 55°.

Hence, the value of x = 55 and y = 55.