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Thm. 12.4: 1.) Congruent central angles have congruent chords (vice versa) 2.) Congruent chords have congruent arcs 3.) Congruent arcs have congruent central angles
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NO WARMUP Copy thm 12.9, corollaries, and thm
12.10 from pgs 679 and 680
12.2/12.3: CHORDS AND
ARCS & INSCRIBED
ANGLESLEQ: WHAT ARE THE THEOREMS
INVOLVED AND CALCULATIONS WITH CHORDS AND ARCS?
THEOREM SHEETThm. 12.4:1.) Congruent central angles have congruent chords
(vice versa)
2.) Congruent chords have congruent arcs
3.) Congruent arcs have congruent central angles
NOTES: PROOF OF THM. 12.4In the circle on the right, prove if
m<CAD=m<FAE, the CD=FR. C
E
F
D
A
THEOREM SHEET
Ex. 3: Find AB.
All go hand in hand:Perp, bisect, and diameter: one makes all the others true.
Ex. 4 P and Q are points on O. The distance from O to PQ is 15 in., and PQ = 16 in. Find the radius of O.
..
More examples: Find the missing lengths.
VOCAB: INSCRIBED ANGLE-an angle whose vertex is on a circle and whose
sides are chords
<ABC and <DEF are inscribed angles in the circles shown below:
<ABC intercepts minor arc AC <DEF intercepts major arc DGF
*intercept means “forms”
THEOREM 12-9: INSCRIBED ANGLE THEOREM The measure of
an inscribed angle is equal to half of its intercepted arcs.
CAmBm
21
B
A
C
EXAMPLES
Find m<ABC Find m<ABC and m<ABD
EXAMPLES
Find m<DEF and mAEC Find m<ABC
3 INSCRIBED < COROLLARIES
If two inscribed angles
intercept the same arc, then the
angles are congruent.
An angle inscribed in a semicircle is a right angle.
If a quadrilateral is inscribed in a circle, then its opposite angles are
supplementary.
THEOREM 12-10The measure of an angle formed by a chord and a
tangent is equal to half the measure of the intercepted arc.
B
D
C C
BD
mBDCCm21
EX: FIND THE NUMBERED ANGLES.
1 2
108˚
32˚
1
161˚
12102˚
46˚
HWK: Copy thms 12.11 and 12.12