Inscribed Angles Chapter 10-4

Find measures of inscribed angles. Find measures of angles of inscribed polygons. intercepted Standard 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. (Key) Standard 21.0 Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles. (Key) Lesson 4 MI/Vocab

Inscribed Angles Def: An angle whose vertex is on a circle and whose sides contain chords The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the Intercepted Arc of the angle

Inscribed Angles X = ½ Y Y = 2X The measure of an inscribed angle is one half of its intercepted arc. X Y X = ½ Y Y = 2X

Measures of Inscribed Angles 20 40 20 + 40 + a + 108 + a = 360 2a + 168 = 360 2a = 192 a = 96 a 108 Lesson 4 Ex1

Measures of Inscribed Angles 20 40 m1 = m2 = m3 = m4 = m5 = 96 96 180 – (20 + 54) = 106 108 Lesson 4 Ex1

A. 30 B. 60 C. 15 D. 120 A B C D Lesson 4 CYP1

A. 110 B. 55 C. 125 D. 27.5 A B C D Lesson 4 CYP1

A. 30 B. 80 C. 40 D. 10 A B C D Lesson 4 CYP1

A. 110 B. 55 C. 125 D. 27.5 A B C D Lesson 4 CYP1

A. 110 B. 55 C. 125 D. 27.5 A B C D Lesson 4 CYP1

Inscribed Angles mX = m Y If two inscribed angles of a circle intercept the same arc (or  arcs), then the angles are . X Y mX = m Y

Proof with Inscribed Angles Given: Prove: ΔPJK  ΔEHG

Proof with Inscribed Angles 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. ΔPJK  ΔEHG 6. AAS Lesson 4 Ex2

Choose the best reason to complete the following proof. Given: Prove: ΔCEM  ΔHJM Lesson 4 CYP2

3. Vertical angles are congruent. 4. Radii of a circle are congruent. 1. Given 2. ______ 3. Vertical angles are congruent. 4. Radii of a circle are congruent. 5. ASA Proof: Statements Reasons 1. 2. 3. 4. 5. ΔCEM  ΔHJM Alternate Interior Angle Theorem Substitution C. Definition of  angles D. Inscribed angles of  arcs are . Lesson 4 CYP2

9th and 10th grade teachers should cover this topic briefly. Inscribed Arcs and Probability This Topic will not be on the chapter 10 test. But Geometric probability will appear on the CST. 9th and 10th grade teachers should cover this topic briefly. Lesson 4 Ex3

9th and 10th grade teachers should cover this topic briefly. Inscribed Arcs and Probability This Topic will not be on the chapter 10 test. But Geometric probability will appear on the CST. 9th and 10th grade teachers should cover this topic briefly. The probability that is the same as the probability of L being contained in . Lesson 4 Ex3

9th and 10th grade teachers should cover this topic briefly. This Topic will not be on the chapter 10 test. But Geometric probability will appear on the CST. 9th and 10th grade teachers should cover this topic briefly. A. B. C. D. A B C D Lesson 4 CYP3

Inscribed Angles A 90o inscribed angle intercepts a 180o arc C is a right   AB is a diameter of the circle C B A R = 2.5 5 3 4

Lesson 4 TH3

Lesson 4 Ex4

A. 45 B. 90 C. 180 D. 80 A B C D Lesson 4 CYP4

A. 17 B. 76 C. 60 D. 42 A B C D Lesson 4 CYP4

A. 17 B. 76 C. 60 D. 42 A B C D Lesson 4 CYP4

A. 73 B. 30 C. 60 D. 48 A B C D Lesson 4 CYP4

Inscribed Polygons Reminder: If all the vertices of a polygon lie on the circle The polygon is inscribed in the circle The circle is circumscribed about the polygon A quadrilateral can be inscribed in a circle  its opposite angles are supplementary Supplementary—adds up to 180o

3x + 2 2x-7 Inscribed Polygons (3x + 2) + (2x –7) =180 5x – 5 = 180

Angles of an Inscribed Quadrilateral Draw a sketch of this situation. Lesson 4 Ex5

Angles of an Inscribed Quadrilateral To find we need to know To find first find Inscribed Angle Theorem Sum of arcs in circle = 360 Subtract 174 from each side. Lesson 4 Ex5

Angles of an Inscribed Quadrilateral Inscribed Angle Theorem Substitution Divide each side by 2. Since we now know three angles of a quadrilateral, we can easily find the fourth. mQ + mR + mS + mT = 360 360° in a quadrilateral 87 + 102 + 93 + mT = 360 Substitution mT = 78 Subtraction Answer: mS = 93; mT = 78 Lesson 4 Ex5

A. 99 B. 104 C. 81 D. 76 A B C D Lesson 4 CYP5

A. 99 B. 104 C. 81 D. 76 A B C D Lesson 4 CYP5

Homework Chapter 10-4 Pg 583 6 – 8, 15 – 21, 24 – 31 all