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Transcript

10.4 Inscribed Angles Objectives Find measures of inscribed angles Find measures of angles of inscribed polygons Inscribed Angles An inscribed angle is an angle that has its vertex on the circle and its sides are chords of the circle. A C B Inscribed Angles Theorem 10.5 A (Inscribed Angle Theorem): The measure of an inscribed angle equals ½ the measure of its intercepted arc (or the measure of the intercepted arc is twice the measure of the inscribed angle). C B mACB = ½m 2 mACB = or Example 1: In and Find the measures of the numbered angles. Example 1: First determine Arc Addition Theorem Simplify. Subtract 168 from each side. Divide each side by 2. Example 1: So, m Example 1: Answer: Your Turn: In and measures of the numbered angles. Answer: Find the Inscribed Angles Theorem 10.6: If two inscribed s intercept arcs or the same arc, then the s are . mDAC mCBD Example 2: Given: Prove: Example 2: Proof: Statements Reasons 1. 1. Given 2. 2. If 2 chords are , corr. minor arcs are . 3. 3. Definition of intercepted arc 4. 4. Inscribed angles of arcs are . 5. 5. Right angles are congruent 6. 6. AAS Your Turn: Given: Prove: Your Turn: Proof: Statements Reasons 1. 1. Given 2. 2. Inscribed angles of arcs are . 3. 3. Vertical angles are congruent. 4. 4. Radii of a circle are congruent. 5. 5. ASA Example 3: PROBABILITY Points M and N are on a circle so that . Suppose point L is randomly located on the same circle so that it does not coincide with M or N. What is the probability that Since the angle measure is twice the arc measure, inscribed must intercept , so L must lie on minor arc MN. Draw a figure and label any information you know. Example 3: The probability that is the same as the probability of L being contained in . Answer: The probability that L is located on is Your Turn: PROBABILITY Points A and X are on a circle so that Suppose point B is randomly located on the same circle so that it does not coincide with A or X. What is the probability that Answer: Angles of Inscribed Polygons Theorem 10.7: If an inscribed intercepts a semicircle, then the is a right . i.e. If AC is a diameter of , then the mABC = 90°. o Angles of Inscribed Polygons Theorem 10.8: If a quadrilateral is inscribed in a , then its opposite s are D supplementary. A B O i.e. Quadrilateral ABCD is inscribed in O, thus A and C are supplementary and B and D are supplementary. C Example 4: ALGEBRA Triangles TVU and TSU are inscribed in with Find the measure of each numbered angle if and Example 4: are right triangles. since they intercept congruent arcs. Then the third angles of the triangles are also congruent, so . Angle Sum Theorem Simplify. Subtract 105 from each side. Divide each side by 3. Example 4: Use the value of x to find the measures of Given Answer: Given Your Turn: ALGEBRA Triangles MNO and MPO are inscribed in with Find the measure of each numbered angle if and Answer: Example 5: Quadrilateral QRST is inscribed in find and Draw a sketch of this situation. If and Example 5: To find To find we need to know first find Inscribed Angle Theorem Sum of angles in circle = 360 Subtract 174 from each side. Example 5: Inscribed Angle Theorem Substitution Divide each side by 2. To find find we need to know but first we must Inscribed Angle Theorem Example 5: Sum of angles in circle = 360 Subtract 204 from each side. Inscribed Angle Theorem Divide each side by 2. Answer: Your Turn: Quadrilateral BCDE is inscribed in find and Answer: If and Assignment Geometry Pg. 549 #8 – 10, 13 – 16, 18 – 20 Pre-AP Geometry Pg. 549 #8 – 10, 13 – 20