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3 Inscribed AnglesCHAPTER1 Inscribed AnglesLearning Objectives Find the measure of Inscribed Angles and the arcs they QueueWe are going to use #14 from the homework in the previous What is the measure of each angle in the triangle? How do you know?2. What do you know about the three arcs?3. What is the measure of each arc?4. What is the relationship between the Angles in the triangles and the measure of each arc?Know What?Your family went to Washington DC over the summer and saw the White House.

4 The closest you canget to the White House are the walking trails on the far right. You got as close as you could (on the trail) to the fenceto take a picture (you were not allowed to walk on the grass). Where else could you have taken your picture from toget the same frame of the White House? Where do you think the best place to stand would be?Your line of sight inthe camera is marked in the picture as the grey lines. The white dotted arcs do not actually exist, but were added tohelp with this AnglesWe have discussed central Angles so far in this chapter.

5 We will now introduce another type of angle, the Angle:An angle with its vertex is the circle and its sides contain Arc:The arc that is on the interior of the Inscribed angle and whose endpoints are on the vertex of an Inscribed angle can be anywhere on the circle as long as its sides intersect the circle to form anintercepted , we will investigation the relationship between the Inscribed angle, the central angle and the arc they 9-4: Measuring an Inscribed AngleTools Needed: pencil, paper, compass, ruler, protractor1.

6 Draw three circles with three different Inscribed Angles . For A, make one side of the Inscribed angle a diameter,for B, makeBinside the angle and for CmakeCoutside the angle. Try to make all the Angles different Using your ruler, draw in the corresponding central angle for each angle and label each set of 1. Inscribed Angles3. Using your protractor measure the six Angles and determine if there is a relationship between the central angle,the Inscribed angle, and the intercepted LM=m NP=m QR=m6 LKM=m6 NOP=m6 QSR= Inscribed Angle Theorem:The measure of an Inscribed angle is half the measure of its intercepted the picture,m6 ADC=12m AC.

7 If we had drawn in the central angle6 ABC, we could also say thatm6 ADC=12m6 ABCbecause the measure of the central angle is equal to the measure of the intercepted prove the Inscribed Angle Theorem, you would need to split it up into three cases, like the three different anglesdrawn from Investigation 9-4. We will touch on the algebraic proofs in the review 1:Findm :From the Inscribed Angle Theorem,m DC=2 45 =90 .m6 ADB=12 76 =38 .Example 2 :The intercepted arc for both Angles is AB. Therefore,m6 ADB=m6 ACB=12 124 =62 This example leads us to our next 9-8: Inscribed Angles that intercept the same arc are prove Theorem 9-8, you would use the similar triangles that are formed by the 3:Findm6 DABin :BecauseCis the center,DBis a diameter.

8 Therefore,6 DABinscribes semicircle, or 180 .m6 DAB=12 180 =90 .Theorem 9-9:An angle that intercepts a semicircle is a right Theorem 9-9 we could also say that the angle is Inscribed in a semicircle. Anytime a right angle is Inscribed ina circle, the endpoints of the angle are the endpoints of a diameter. Therefore, the converse of Theorem 9-9 is the three vertices of a triangle are on the circle, like in Example 3, we say that the triangle isinscribedin thecircle. We can also say that the circle iscircumscribedaround (or about) the triangle.

9 Any polygon can be inscribedin a 4:Findm6 PMN,m PN,m6 MNP,m6 LNP, andm :m6 PMN=m6 PLN=68 by Theorem PN=2 68 =136 from the Inscribed Angle by Theorem 92 =46 from the Inscribed Angle findm LN, we need to the third angle in4 LPN, so 68 +46 +m6 LPN=180 .m6 LPN=66 , which means thatm LN=2 66 =132 . Inscribed QuadrilateralsThe last theorem for this section involves inscribing a quadrilateral in a 1. Inscribed AnglesInscribed Polygon:A polygon where every vertex is on a , that not every quadrilateral or polygon can be Inscribed in a circle.

10 Inscribed quadrilaterals are also calledcyclic these types of quadrilaterals, they must have one special property. We will investigate 9-5: Inscribing QuadrilateralsTools Needed: pencil, paper, compass, ruler, colored pencils, scissors1. Draw a circle. Mark the center Place four points on the circle. Connect them to form a quadrilateral. Color the 4 Angles of the quadrilateral 4different Cut out the quadrilateral. Then cut the quadrilateral into two triangles, by cutting on a Line up6 Band6 Dso that they are adjacent Angles .