Geometry Inscribed Angles

August 24, 2015 Goals  Know what an inscribed angle is.  Find the measure of an inscribed angle.  Solve problems using inscribed angle theorems.

August 24, 2015 Inscribed Angle The vertex is on the circle and the sides contain chords of the circle. A C B ABC is an inscribed angle. AC is the intercepted arc.

August 24, 2015 Inscribed Angle A C B How does mABC compare to m AC?

August 24, 2015 Draw circle O, and points A & B on the circle. Draw diameter BR. O B A R

August 24, 2015 Draw radius OA and chord AR. O B A R 1 2 3

August 24, 2015 (Very old) Review  The Exterior Angle Theorem (4.2)  The measure of an exterior angle of a triangle is equal to the sum of the two remote, interior angles m1 + m2 = m3

August 24, 2015 mARO + mOAR = mAOB O B A R What type of triangle is  OAR? Isosceles The base angles of an isosceles triangle are congruent. 1  

August 24, 2015 mARO + mOAR = mAOB O B A R m1 + m2 = m3 But m1 = m2 m1 + m1 = m3 2m1 = m3 m1 = (½)m3 This angle is half the measure of this angle

August 24, 2015 Where we are now. O B A R xx (x/2) Recall: the measure of a central angle is equal to the measure of the intercepted arc. xx m  1 = (½)m 

August 24, 2015 Theorem 12.8 O B A R (x/2) If an angle is inscribed in a circle, then its measure is one- half the measure of the intercepted arc. xx Inscribed Angle Demo

August 24, 2015 Example 1 88 ? 44

August 24, 2015 Example 2 A B C 85 ? 170

August 24, 2015 Example 3 xx 200 100 The circle contains 360. 360 – ( ) = 60 30 ? 60

August 24, 2015 Another Theorem 2x xx xx ? ? Theorem 10.9 If two inscribed angles intercept the same (or congruent) arcs, then the angles are congruent. Theorem Demonstration

August 24, 2015 A very useful theorem. Draw a circle. Draw a diameter. Draw an inscribed angle, with the sides intersecting the endpoints of the diameter.

August 24, 2015 A very useful theorem. What is the measure of each semicircle? 180 What is the measure of the inscribed angle? 90

August 24, 2015 Theorem If an angle is inscribed in a semicircle, then it is a right angle. Theorem Demo

8/24/2015 Theorem 12.2: Tangent-Chord A B C 12 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 the intercepted arc.

8/24/2015 Simplified Formula aa bb 1 2

8/24/2015 Example 1 A B C 80 160  200 

8/24/2015 Example 2. Solve for x. A B C 4x (10x – 60)

August 24, 2015 Inscribed Polygon  The vertices are all on the same circle.  The polygon is inside the circle; it is inscribed.

August 24, 2015

A cyclic quadrilateral has all of its vertices on the circle. B A C D

August 24, 2015 An interesting theorem. A B C D

August 24, 2015 An interesting theorem. A B C D

August 24, 2015 An interesting theorem. A B C D Adding the equations together…

August 24, 2015 An interesting theorem. A B C D

August 24, 2015 An interesting theorem.

August 24, 2015 An interesting theorem. A B C D BAD and BCD are supplementary.

August 24, 2015 Theorem A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. m 1 + m 3 = 180 & m 2 + m 4 = 180 Theorem Demo

August 24, 2015 Example Solve for x and y. 4x 2x 5y 100 4x + 2x = 180 6x = 180 x= 30 and 5y = 180 5y = 80 y = 16

August 24, 2015 Summary  The measure of an inscribed angle is one-half the measure of the intercepted arc.  If two angles intercept the same arc, then the angles are congruent.  The opposite angles of an inscribed quadrilateral are supplementary.

August 24, 2015 Practice Problems Inscribed Hexagon