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. C A B

Inscribed Angles Theorem 10.5 (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 A B mACB = ½m or 2 mACB =

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 Find the measures of the numbered angles. Answer:

Inscribed Angles Theorem 10.6: If two inscribed s intercept  arcs or the same arc, then the s are . mDAC  mCBD

Example 2: Given: Prove:

Example 2: Proof: Statements Reasons 1. Given 1. 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: 1. Given 2. Inscribed angles of arcs are . 3. Vertical angles are congruent. 4. Radii of a circle are congruent. 5. ASA Proof: Statements Reasons 1. 2. 3. 4. 5.

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 mABC = 90°. o

Angles of Inscribed Polygons Theorem 10.8: If a quadrilateral is inscribed in a , then its opposite s are supplementary. i.e. Quadrilateral ABCD is inscribed in O, thus A and C are supplementary and B and D are supplementary. D A C B O

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 Given Answer:

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 If and find and Draw a sketch of this situation.

Example 5: To find we need to know To find 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 we need to know but first we must find 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 If and find and Answer:

Assignment Geometry Pg. 549 #8 – 10, 13 – 16, 18 – 20 Pre-AP Geometry Pg. 549 #8 – 10, 13 – 20