Cyclic Quadrilaterals

In the given figure, $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and $\mathrm{D}$ are four points on a circle. $\mathrm{A}\mathrm{C}$ and $\mathrm{B}\mathrm{D}$ intersect at a point $\mathrm{E}$ such that $\mathrm{\angle }\mathrm{B}\mathrm{E}\mathrm{C}=130°$ and $\mathrm{\angle }\mathrm{E}\mathrm{C}\mathrm{D}=20°.$ If $\mathrm{\angle }\mathrm{B}\mathrm{A}\mathrm{C}=\mathrm{k}°$, then find the value of $\mathrm{k}$.

In the given figure, $\mathrm{\angle }\mathrm{A}\mathrm{B}\mathrm{C}=69°, \mathrm{\angle }\mathrm{A}\mathrm{C}\mathrm{B}=31°$, find $\mathrm{\angle }\mathrm{B}\mathrm{D}\mathrm{C}.${write only numerical value without degrees symbol}

In the given figure, $\mathrm{\angle }\mathrm{P}\mathrm{Q}\mathrm{R}=100°,$ where $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ are points on a circle with centre $\mathrm{O}.$ Find $\mathrm{\angle }\mathrm{O}\mathrm{P}\mathrm{R}$.

Prove that a cyclic parallelogram is a rectangle.

$\mathrm{A}\mathrm{B}\mathrm{C}$ and $\mathrm{A}\mathrm{D}\mathrm{C}$ are two right triangles with common hypotenuse $\mathrm{A}\mathrm{C}$. Prove that $\mathrm{\angle }\mathrm{C}\mathrm{A}\mathrm{D}=\mathrm{\angle }\mathrm{C}\mathrm{B}\mathrm{D}$.

If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.

Two circles intersect at two points $\mathrm{B}$ and $\mathrm{C}.$ Through $\mathrm{B},$ two line segments $\mathrm{A}\mathrm{B}\mathrm{D}$ and $\mathrm{P}\mathrm{B}\mathrm{Q}$ are drawn to intersect the circles at $\mathrm{A}, \mathrm{D}$ and $\mathrm{P}, \mathrm{Q}$ respectively. Prove that $\mathrm{\angle }\mathrm{A}\mathrm{C}\mathrm{P}=\mathrm{\angle }\mathrm{Q}\mathrm{C}\mathrm{D}.$

If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

$\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a cyclic quadrilateral whose diagonals intersect at a point $\mathrm{E}.$ If $\mathrm{\angle }\mathrm{D}\mathrm{B}\mathrm{C}=70°, \mathrm{\angle }\mathrm{B}\mathrm{A}\mathrm{C}$ is $30°,$ find $\mathrm{\angle }\mathrm{B}\mathrm{C}\mathrm{D}.$ Further, if $\mathrm{A}\mathrm{B}=\mathrm{B}\mathrm{C},$ find $\mathrm{\angle }\mathrm{E}\mathrm{C}\mathrm{D}.$

A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

In the given figure, $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are three points on a circle with a centre $\mathrm{O}$ such that $\mathrm{\angle }\mathrm{B}\mathrm{O}\mathrm{C}=30°$ and $\mathrm{\angle }\mathrm{A}\mathrm{O}\mathrm{B}=60°$. If $\mathrm{D}$ is a point on the circle other than the arc$\mathrm{A}\mathrm{B}\mathrm{C},$ find $\mathrm{\angle }\mathrm{A}\mathrm{D}\mathrm{C}.$