Angles and Arcs

When two radii of a circle are not in a role, then central angles and arcs are formed. A central angle of a circle is an angle with measure less than 180 whose vertex is the center of the circle. An arc is the set of points that lies on the circle with an angle measure of the central angle that is corresponding to the arc. There are two kinds of arc. The first kind is called the minor arc. A minor arc is corresponding to the central angle. The other kind of angle is called the major arc. A major arc is corresponding to the angle with an angle measure of (360-the central angle). Followings are some additional concepts about the central angles, arcs, and their relations:

Arc Addition Postulate: If P is a point on arc APB, then arcs AP + arcs PB = arcs APB.

Concentric circles are coplanar circles with a common center.

In the same circle or congruent circles, congruent arcs are arcs that have the same degree measure.

In the same circle or in congruent circles:

If chords are congruent, then their corresponding arcs and central angles are congruent;
If arcs are congruent, then their corresponding chords and central angles are congruent;
If central angles are congruent, then their corresponding arcs and chord s are congruent.
When you move the vertex of the central angle to a point on the circle and extend the two legs of the central angle to the points on the circle, you get an inscribed angle. Here are some concepts about The Inscribed Angle:

The Inscribed Angle Theorem: The measure of an inscribed angle is one-half of the degree measure of its intercepted arc.

If two inscribed angles intercept the same arc or congruent arcs, then the angles are congruent.

An angle inscribed in a semicircle is a right angle.

If two arcs of a circle are included between parallel chords or secants, then the arcs are congruent.

The opposite angles of an inscribed quadrilateral are supplementary.

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