M L Aggarwal Solutions for Chapter: Circles, Exercise 2: Exercise 10.2

The angles subtended by a chord at any two points of a circle are equal.

$AOB$ is a diameter of a circle and $C$ is any point on the circle then $A{C}^2+B{C}^2=A{B}^2$

If $A, B, C$ and $D$ are four points such that $\angle BAC=30°$ and $\angle BDC=60°$, then $D$ is the centre of the circle through $A, B$ and $C$.

In the figure below, $O$ is the centre of the circle. If $\angle AOC=130°$, find the value of $x$ in degrees.

In the figure below, $AB$ is the diameter of the circle. If $\angle ACD=35°$, find the value of $x$ in degrees

In the adjoining figure, two circles with centres $O$, $C$ intersect at points $A,B$. The centre $C$ of the smaller circle lies on the circumference of the other circle. If $\angle \mathrm{AOB}=80°$, find $\angle \mathrm{APB}.$

$O$ is the circumcentre of the triangle $ABC$ and $D$ is mid-point of the base $BC$. Prove that $\angle BOD=\angle \mathrm{A}.$

In the adjoining figure, $AB$ and $CD$ are equal chords. $AD$ and $BC$ intersect at $E$. Prove that $\mathrm{AE}=\mathrm{CE}$ and $\mathrm{BE}=\mathrm{DE}$