Download 14.1 Inscribed Angles

Geometry – Chapter 14 Lesson Plans
Section 14.1 – Inscribed Angles
Enduring Understandings: The student shall be able to:
1. Identify and use properties of inscribed angles.
Standards:
30. Circles
Identifies and defines circles and their parts (center, arc, interior, exterior); segments
and lines associated with circles (chord, diameter, radius, tangent, secant); properties
of circles (congruent, concentric, tangent); relationship of polygons and circles
(inscribed, circumscribed); angles (central; inscribed; formed by tangents, chords, and
secants).
Essential Questions: How can we find the measure of an inscribed angle?
Warm up/Opener:
Activities:
Review central angles – an angle whose vertex is at the center of the circle.
We are now going to talk about inscribed angles. Think inscribed polygons – iff every
vertex of the polygon lies on the circle.
Defn: An angle is inscribed iff its vertex lies on the circle and its sides contain chords of
the circle.
Show some examples of inscribed angles and non-inscribed angles (one line being a
tangent). What are the intercepted arcs?
Thm. 14-1: The degree measure of an inscribed angle equals one-half the degree measure
if its intercepted arc.
Proof?
1.
2.
3.
A
This proof is done in three parts:
With the side of the inscribed angle including the center of the circle,
With the center of the circle inside the inscribed angle, and
With the center outside of the inscribed angle.
B
1
A
B
2
C
A
O
3
C
A
B
C
1. Let AC be the diameter of circle O.
Then BOC is a central angle.
AOB is an isosceles triangle, so A  B
BOC is an external angle to AOB, so  BOC = A + B
By substitution,  BOC = 2 * A, or A = ½  BOC
2. Make a more general statement with center of the circle inside AB and AC. By
the angle addition postulate, it can be shown the proof is still valid.
3. The last case being if the center is outside of AB and AC. Again, by the angle
addition postulate, it can be shown the proof is still valid.
This is a powerful theorem!
Do examples with one angle, then an example giving the central angle and calculating the
inscribed angles, leading to the following theorem:
B
C
A
D
m BC = 88
Find m A and
D
Thm. 14-2: If inscribed angles intercept the same arc or congruent arcs, then the angles
are congruent.
And the special case:
Thm. 14-3: If an inscribed angle of a circle intercepts a semicircle, then the angle is a
right angle.
Do example 2 of the Red Book, page 468.
Do example 3 of the Red Book, page 469, which leads us to the following theorem:
Additional Thm: If a quadrilateral is inscribed in a circle, then its opposite angles are
supplementary.
B
A
C
D
Proof:
 Due to limitations of Word, “m BD” means the measure of arc BD in degrees, and
“m BCD” means the measure of arc BCD in degrees.
 Angle A and Angle C are opposite.
 Angle C inscribes the minor arc BD, and Angle A inscribes the major arc BCD.
 The m  A = ½ m BCD = ½ (360 – m BD).
 The m  C = ½ m  BD.
 Adding Angle A to Angle C gives:
m  A + m  C = ½ (360 – m BD) + ½ m  BD = ½ * 360 = 180
Assessments:
Do the “Check for Understanding” 2-7
CW WS 14.1
HW pg 590 - 591, # 9 - 32 all (24) if I only do this section
HW pg 590 – 591, # 9 – 31 odd (12) if I combine with another section
Extra Credit: Enrichment 14.1