Circle theorems Questionbank

A tangent is drawn to a circle at point $A$. A chord $AB$ is drawn such that it is parallel to another chord $BC$ (where $C$ is on the circumference). If $\angle ABC = 55^\circ$, find the size of $\angle BAC$.

A cyclic quadrilateral ${ABCD}$ is inscribed in a circle. If the side ${AD}$ is extended to a point ${E}$ outside the circle, which of the following statements is always true regarding the exterior angle ${\angle CDE}$?

Points $A$, $B$, and $C$ lie on a circle with center $O$. If the angle at the circumference is $\angle ABC = 38^\circ$, what is the size of the central angle $\angle AOC$ standing on the same arc $AC$?

Two tangents from an external point $T$ touch a circle of radius $r$ at points $A$ and $B$. If the distance from the centre $O$ to point $T$ is $2r$, what is the size of $\angle ATB$?

The following diagram shows a circle with center $O$ and diameter $AB$. Point $C$ lies on the circumference.

The length of the chord $AC$ is $5$ units and the angle $\angle BAC = 60^\circ$.

Find the length of the diameter $AB$.

In a cyclic quadrilateral $ABCD$, the interior angle $A\hat{B}C = x$ and the opposite interior angle $A\hat{D}C = 2x - 30^\circ$.

Calculate the value of $x$.

In a cyclic quadrilateral $ABCD$, it is given that the side lengths $AB$ and $AD$ are equal. If $\angle BCD = 80^\circ$, calculate the size of the angle $\angle ABD$.

The specific point where a tangent line meets the circumference of a circle without crossing into the interior is formally known as the:

A chord divides a circle into two distinct regions. What is the name specifically given to the region bounded by the chord and the major arc?

Fill in the blank: According to the central angle theorem, the angle subtended by an arc at the center is exactly ______ the angle subtended by the same arc at any point on the remaining part of the circumference.