Day 4 Inscribed Angles
ACTIVITY:
Investigating Inscribed Angles
Question:
An angle in a circle is an inscribed
angle if its vertex is on the circle and its sides contain chords of the
circle. How is the measure of an
inscribed angle related to the measure of the corresponding central angle?
Exploring
the Concept:
1. Construct a circle. Label its center P.
2. Use a straightedge to
construct a central angle. Label it 
.
3. Locate three points on
Circle P in the exterior of and label them T, U, and V. Use a straightedge to draw the inscribed
angles 
.
, and
.
Investigate:
4. Use a protractor to measure , , , and. Make a table similar
to the one below. Record the angle measures
for Circle 1 in the table.
5. Repeat Steps 1 through 3
using different central angles. Record
the measures in your table.
Make
a Conjecture:
6. Use the results in your
table to make a conjecture about how the measure of an inscribed angle is related
to the measure of the corresponding central angle.
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.
Measure
of an Inscribed Angle Theorem
If
an angle is inscribed in a circle, then its measure is half the measure of its
intercepted arc.
This
proof can be split into 3 different cases. Each will differ in where the center
lies in relation to the inscribed angle.
Proof:
Theorem: If two inscribed angles of a circle
intercept the same arc, then the angles are congruent.
Proof:
