Download Chapter 10.4 Notes: Use Inscribed Angles and Polygons

Chapter 10.4 Notes: Use
Inscribed Angles and Polygons
Goal: You will use inscribed angles of circles.
• An inscribed angle is an angle whose vertex is on a
circle and whose sides contain chords of the circle.
• The arc that lies in the interior of an inscribed angle
and has endpoints on the angle is called the
intercepted arc of the angle.
• Theorem 10.7 Measure of an Inscribed Angle
Theorem:
The measure of an inscribed angle is one half the
measure of its intercepted arc.
Ex.1: Find the indicated measure in
a. mT
P.
b. mQR
Ex.2: Find mRS and mSTR. What do you notice
about STR and RUS ?
• Theorem 10.8:
If two inscribed angles of a circle intercept the same
arc, then the angles are congruent.
Ex.3: Name two pairs of congruent angles in the
figure.
• Find the measure of the red arc or angle.
Ex.4:
Ex.5:
Ex.6:
Polygons
• A polygon is an inscribed polygon if all of its
vertices lie on a circle.
• The circle that contains the vertices is a
circumscribed circle.
• Theorem 10.9:
If a right triangle is inscribed in a circle, then the
hypotenuse is a 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 the right angle.
• Theorem 10.10:
A quadrilateral can be inscribed in a circle if and
only if its opposite angles are supplementary.
• Find the value of each variable.
Ex.7:
Ex.8:
• Find the value of each variable.
Ex.9:
Ex.10:
Ex.11: Find the indicated measure in
a. mD
C.
b. m AB
Ex.12: Find mKN and mKMN. What do you
notice about KMN and KLN ?