Inscribed Angles

May 13, 2008

What is an inscribed angle?• An inscribed angle

is an angle whose vertex is on a circle and whose sides contain chords of the circle. We say that angle 1 above intercepts the arc shown in red.

Measure of an inscribed angle

• Theorem: The measure of an inscribed angle is equal to half the measure of its intercepted arc.

More about inscribed angles

Proving the measure of an inscribe angle

• Try this one…

B

A

C

D

1st Corollary

• Corollary 1: If two inscribed angles intercept the same arc, then the angles are congruent.

2nd Corollary

• Corollary 2: An angle inscribed in a semicircle is a right angle.

3rd Corollary

• Corollary 3: If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

Angles formed by a chord and a tangent

• Theorem: The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc.

More about chords

• Theorem: The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs.

Secants and tangent• Theorem: The measure of an angle formed

by two secants, two tangents, or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs.

Two secants

Two tangents..

A tangent and a secant