Download Sect. 12-4 Inscribed Angles

Sect. 12-4
Inscribed 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 ∠𝐶 is on the circle, and the sides
of ∠𝐶 are chords of the circle.
• ∠𝐶 is an inscribed angle. 𝐴𝐵 is the
intercepted arc of ∠C.
Measuring Inscribed Angles
• A polygon is inscribed in a circle if all its
vertices lie on the circle.
– ∆𝐷𝐸𝐹 is inscribed in circle Q.
– Circle Q is circumscribed
about ∆𝐷𝐸𝐹
Example
• Which arc does ∠𝐴 intercept?
• Which angle intercepts 𝐷𝐴𝐵?
• Is quadrilateral ABCD inscribed in the circle or
is the circle inscribed in ABCD?
Theorem 12-10
Inscribed Angle Theorem
• The measure of an inscribed angle is half the
measure of its intercepted arc.
• 𝑚∠𝐵 =
1
𝑚𝐴𝐶
2
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, 𝑚∠𝐴 remains the same, and that
1
𝑚∠𝐴 = 𝑚𝐵𝐶. This is also true when A and C
2
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
• 𝐾𝐽 is tangent to the circle at J. Find the values
of x, y and z.