# Intercepted Arc

Intercepted arc is that arc which is formed when segments intersect parts of a circle and create arcs.

The intercepted arc in the figure on the left is $\widehat{EF}$. The angle thus formed at the center is called the Central angle or the intercepted angle.

The measure of a central angle in a circle is always equal to the measure of its intercepted arc.

m $\angle$EOF = m$\widehat{EF}$

When two straight lines intersect a circle, that part of the circle between the intersection points is called the intercepted arc. The lines intercept, or 'cut off', the arc of the circle.

Usually, the two lines become the arms of an angle, as shown in the intercepted arcs figure above and such angle is called the intercepted angle. But this is not always the case. For example, in the figure on the left, the two secant lines cut off, or intercept two arcs, $\widehat{CE}$ and $\widehat{BD}$.

In this case the the vertex (at P in the figure on the left) of the angle lies outside the circle. The measure of an angle with its vertex outside the circle is half the difference of the measures of the intercepted arcs.

The formula is, Angle = (measure of difference of intercepted arcs)**The formula is** m$\angle$BPD = m($\widehat{BD}$)- m($\widehat{CE}$)

In this case the two lines do not form an angle at the center but become the arms of an angle whose vertex is on the circle. This angle is called the inscribed angle. The sides of this angle (or the arms) become the chords of the circle. They form an intercepted arc BD.

An inscribed angle always has its vertex on the circle. The measure of an inscribed angle equals $\frac{1}{2}$ the measure of the intercepted arc.**The formula is** m$\angle$BCD = $\frac{1}{2}$ (m($\widehat{BD}$)

In this case the two lines do not form an angle at the center but part of the lines become the arms of an angle whose vertex is inside the circle. They form intercepted arcs EF and GH.

The measure of an angle with its vertex (at P in the figure on the left) inside the circle is half the sum of the intercepted arcs.**The formula is** m$\angle$EPF = m$\angle$GPH = $\frac{1}{2}$(m($\widehat{EF}$) +m($\widehat{GH}$))