sect. 12-4 inscribed angles geometry honors. what and why what? – find the measure of inscribed...
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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.