Section 10.3 Inscribed Angles

Section 10.3 – Inscribed Angles

An inscribed angle is an angle with vertex on a circle and sides that contain the chords of the circle. In the figure, ÐBAC is an inscribed angle and BC is its intercepted arc.

Theorem 10.8 – If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.

mÐABC = ½mAB

Theorem 10.9 – If two inscribed angles intercept the same arc, then the angles are congruent.

mÐABC = mÐADB

Inscribed polygons – polygons are inscribed if all its vertices are on the circle.

Circumscribed polygons – polygons are circumscribed about a circle if each of its sides is tangent to the circle.

Theorem 10.10 – If a right triangle is inscribed in a circle, then the hypotenuse is the diameter of the circle. Conversely if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is a right angle.

Theorem 10.11 – If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary (i.e., add up to 180 degrees).

mÐA + mÐC = 180 degrees;

mÐB + mÐD = 180 degrees