Circle theorems I：练习题

A circle has a diameter AB. Point C lies on the circumference of the circle. If \(\angle ABC = 35^\circ\), calculate the size of \(\angle BAC\).

Points P, Q, and R lie on a circle with center O. If the angle \(\angle PQR = 115^\circ\), calculate the size of the minor angle \(\angle POR\).

Points A, B, and C lie on the circumference of a circle with center O. If the angle \(\angle ABC = 110^\circ\), calculate the reflex angle \(\angle AOC\).

In a circle, PT is a tangent at point T. A and B are points on the circumference such that AB is a chord. If \(\angle BTP = 48^\circ\) and \(\angle ABT = 72^\circ\), calculate the size of \(\angle BAT\).

In the diagram, points A, B, and C lie on the circumference of a circle with center O. Line AT is a tangent to the circle at point A. If \( \angle OAB = 25^\circ \), calculate the size of angle \( \angle BAT \).

In the circle shown, the chord AB makes an angle of \( 58^\circ \) with the tangent to the circle at point A. Point C is located on the major arc AB. Using the appropriate circle theorem, find the size of angle \( \angle ACB \).

Points A, B, and C lie on a circle with center O. AB is a diameter of the circle. A tangent is drawn at point C, and a point T lies on this tangent. If \(\angle ABC = 32^\circ\), calculate the size of the angle between the tangent CT and the chord AC.

In the diagram, A, B, C, and D are points on the circumference of a circle with centre O.
AC is a diameter of the circle and AD is parallel to BC.
The tangent to the circle at point C meets the line AB extended at point T.

(a) If \(\angle ACB = 32^\circ\), find the size of \(\angle BAC\), giving a geometric reason for your answer. [2]
(b) Find the size of \(\angle ATC\). [2]
(c) Determine the size of the obtuse angle \(\angle ADC\) and explain your reasoning. [2]