Chords and Inscribed Angles in Circles
Copy thm 12.9, corollaries, and thm 12.10
from pgs 679 and 680
LEQ: WHAT ARE THE THEOREMS
INVOLVED AND CALCULATIONS WITH
CHORDS AND ARCS?
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
In the circle on the right, prove if
m<CAD=m<FAE, the CD=FR.
C
E
F
D
A
Ex. 3: Find AB.
All go hand in hand:
Perp, bisect, and
diameter: one makes all
the others true.
More examples: Find the missing lengths.
-
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”
The measure of
an inscribed angle
is equal to half of
its intercepted
arcs.
B
A
C
Find m<ABC
Find m<ABC and m<ABD
Find m<DEF and mAEC
Find m<ABC
The 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
B
D
1
2
108˚
32˚
1
1
61˚
1
2
102˚
46˚
Copy thms 12.11 and 12.12
Explore the relationships between central angles, chords, arcs, and inscribed angles in circles through theorems and examples. Learn about the Inscribe Angle Theorem and its corollaries to deepen your understanding of circle geometry concepts.
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Presentation Transcript
NO WARMUP Copy thm 12.9, corollaries, and thm 12.10 from pgs 679 and 680
12.2/12.3: CHORDS AND ARCS & INSCRIBED ANGLES LEQ: WHAT ARE THE THEOREMS INVOLVED AND CALCULATIONS WITH CHORDS AND ARCS?
THEOREM SHEET 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
NOTES: PROOF OF THM. 12.4 In the circle on the right, prove if m<CAD=m<FAE, the CD=FR. C D F A E
THEOREM SHEET Ex. 3: Find AB.
All go hand in hand: Perp, bisect, and diameter: one makes all the others true.
P and Q are points on O. The distance from O to PQ is . Ex. 4 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 A The measure of an inscribed angle is equal to half of its intercepted arcs. C B 1 = m B m A C 2
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 12-10 The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc. 1 = m C mBDC 2 B B D D C C
EX: FIND THE NUMBERED ANGLES. 108 1 32 1 2 1 2 102 1 61 46
HWK: Copy thms 12.11 and 12.12
