Sect. 12-4Inscribed Angles

Geometry Honors

What and Why

• What?– Find the measure of inscribed angles and the arcs

they intercept.• Why?– To use the relationships between inscribed angles

and arcs in real-world situations, such as motion pictures.

Recall Central Angle

• A central angle is an angle whose vertex is the center of the circle.

• The arc formed by a central angle is the same measure as the angle.

Inscribed Angles

• The vertex of is on the circle, and the sides of are chords of the circle.

• is an inscribed angle. is the intercepted arc of

Measuring Inscribed Angles

• A polygon is inscribed in a circle if all its vertices lie on the circle. – is inscribed in circle Q.– Circle Q is circumscribed

about

Example• Which arc does intercept?

• Which angle intercepts ?

• Is quadrilateral ABCD inscribed in the circle or is the circle inscribed in ABCD?

Theorem 12-10Inscribed Angle Theorem

• The measure of an inscribed angle is half the measure of its intercepted arc.

There are three cases of this theorem to consider.

Case 1:

• The center is on a side of the angle.

Case 2

• The center is inside the angle.

Case 3

• The center is outside the angle.

Example

• Find the values of a and b in the diagram.

Corollaries

• Corollary 1– Two inscribed angles that intercept the same arc

are congruent.• Corollary 2– An angle inscribed in a semicircle is a right angle.

• Corollary 3– The opposite angles of a quadrilateral inscribed in

a circle are supplementary.

Examples

• Find the measure of the numbered angle.

• In the diagram, B and C are fixed points, and point A moves along the circle. From the Inscribed Angle Theorem, you know that as A moves, remains the same, and that . This is also true when A and C coincide.

Theorem 12-11

• The measure of an angle formed by a chord and a tangent that intersect on a circle is half the measure of the intercepted arc.

Example

• is tangent to the circle at J. Find the values of x, y and z.