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GEOMETRY: Chapter 10
10.3: Inscribed Angles
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.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 672.
Theorem 10.8: Measure of an Inscribed Angle
Theorem
The measure of an inscribed angle is one half
the measure of its intercepted arc.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 672.
Theorem 10.9
If two inscribed angles of a circle intercept the
same arc, then the angles are congruent.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 673.
Ex.1:
Find the indicated measure in
a. mD
C.
b. mAB
Answer: a. 50o; b. 1100
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 673.
Ex. 2:
Find mKN and the mKMN .
What do you notice about KMN and KLN ?
Answer: 104 degrees, 52 degrees. The angles are congruent.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 673.
Ex. 3:
Name two pairs of congruent angles in the figure.
Answer: angle R is congruent to angle S,
Angle U is congruent to angle T
Images taken from: Geometry. McDougal Littell: Boston, 2007. P.674.
Theorem 10.10: 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.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 674.
Ex. 4:A graphic design software program was used
in a home improvement store to design kitchen
cabinets. The designer showed a wall of
cabinets with a 90o viewing angle at P. From
what other positions would the cabinets fill a
90o viewing window?
Answer: from any position on a
semicircle that has segment AB as a diameter
Images taken from: Geometry. McDougal Littell: Boston, 2007. P.674.
Theorem 10.11
A quadrilateral can be inscribed in a circle if and
only if its opposite angles are supplementary.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 675.
Ex. 5:
Find the value of each variable.
Answer: a. x = 80, y = 120; b. x =15, y = 5
Images taken from: Geometry. McDougal Littell: Boston, 2007. P.675.
10.3, p. 616, #3-23 all