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