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Chapter 10.4 & 10.5. 10.4 Using Inscribed Angles and Polygons. Inscribed angle- an angle whose vertex is on a circle and whose sides contain chords of the circle. Intercepted arc- an arc that lies in the interior of an inscribed angle and has endpoints on the angle. - PowerPoint PPT Presentation
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CHAPTER 10.4 & 10.5
10.4 USING INSCRIBED ANGLES AND POLYGONS Inscribed angle- an angle whose vertex is on
a circle and whose sides contain chords of the circle.
Intercepted arc- an arc that lies in the interior of an inscribed angle and has endpoints on the angle.
Intercepted arc
Inscribed angle
Measure of an inscribed angle theorem- The measure of an inscribed angle is one half the measure of its intercepted arc.
mADB = 12 mAB
B
A
CD
Example
Find a. mD =b. mAB =
35 50
A
CDE
B
EXAMPLE
Find a. mDEB =b. mDB =C. DAB
52A
DE
B
THEOREM If 2 inscribed angles of a circle intercept the
same arc, then the angles are congruent.
Inscribed polygon- a polygon in which all the vertices lie on the circle
Circumscribed circle- the circle that contains the vertices of the polygon
THEOREM A quadrilateral can be inscribed in a circle if
and only if its opposite angles are supplementary
mA + mC = 180mB + mD = 180
B
A
DC
EXAMPLE Find x and y.
yx
10060
B
A
DC
EXAMPLE Find x and y.
17y 7x
5x19y
BA
DC
THEOREM 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.
D
A
C
B
10.5 APPLYING OTHER ANGLE RELATIONSHIPS IN CIRCLES Theorem- If a tangent and a chord intersect at a point
on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.
m2 =12 mACB
m1= 12 mAB
2 1A
BC
INTERSECTING LINES AND CIRCLES
outside the circleinside the circleon the circle
ANGLES IN THE CIRCLE THEOREM If the chords intersect inside a circle, then the
measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle
m2 = 12 mAD + mBC
m1 = 12 mDC + mAB
21
A
CB
D
ANGLES OUTSIDE THE CIRCLE THEOREM If a tangent and a secant, 2 tangents, or 2
secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.
321
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