10.4 Use Inscribed Angles 10.4 Use Inscribed Angles and Polygonsand Polygons
Inscribed Angles = ½ the Inscribed Angles = ½ the Measure of the Intercepted Measure of the Intercepted
ArcArc
90
45 45
2 Inscribed Angles 2 Inscribed Angles CorollaryCorollary
• If 2 inscribed angles intercept the same If 2 inscribed angles intercept the same arc, then the angles are congruent.arc, then the angles are congruent.
12
110
m1 = 1 = mm 2 = 55 2 = 55
Inscribed Angle/Semicircle Inscribed Angle/Semicircle CorollaryCorollary
•An angle inscribed in a semicircle is a right An angle inscribed in a semicircle is a right angle.angle.
• Inscribe/Inscribe/CircumscribedCircumscribed
- A circle is - A circle is circumscribed about a circumscribed about a polygonpolygon
and a polygon is and a polygon is inscribed in a circle inscribed in a circle when each vertex of when each vertex of the polygon lies on the the polygon lies on the circle.circle.
Inscribed Quadrilateral Inscribed Quadrilateral CorollaryCorollary
• If a quadrilateral is inscribed in a circle, If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.then its opposite angles are supplementary.
1
32
4
m 1 + 1 + mm 3 = 180 3 = 180
m 2 + 2 + mm 4 = 180 � 4 = 180 �
Chord/Tangent TheoremChord/Tangent Theorem
If a tangent and a chord intersect at a If a tangent and a chord intersect at a point on a circle, then the measure of point on a circle, then the measure of each each formed is ½ the measure of its formed is ½ the measure of its intercepted arc.intercepted arc.
mm1 = ½ 1 = ½ mm AB AB
mm2 = ½ 2 = ½ mm BCA BCA
((((
example:example:Find Find mm1 =1 =
mm BCA = BCA =
mm2 =2 =
7575oo
105105oo
210210oo
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