Five-Minute Check (over Lesson 10–3) Then/Now New Vocabulary Theorem 10.6: Inscribed Angle Theorem Proof: Inscribed Angle Theorem (Case 1) Example 1: Use Inscribed Angles to Find Measures Theorem 10.7 Example 2: Use Inscribed Angles to Find Measures Example 3: Use Inscribed Angles in Proofs Theorem 10.8 Example 4: Find Angle Measures in Inscribed Triangles Theorem 10.9 Example 5: Real-World Example: Find Angle Measures Lesson Menu

A. 60 B. 70 C. 80 D. 90 5-Minute Check 1

A. 40 B. 45 C. 50 D. 55 5-Minute Check 2

A. 40 B. 45 C. 50 D. 55 5-Minute Check 3

A. 40 B. 30 C. 25 D. 22.5 5-Minute Check 4

A. 24.6 B. 26.8 C. 28.4 D. 30.2 5-Minute Check 5

A. B. C. D. 5-Minute Check 6

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10-4 Inscribed Angles (Pg. 709) TARGETS Find measures of inscribed angles. Find measures of angles of inscribed polygons. Then/Now

Content Standards Mathematical Practices 4 Model with mathematics.   G-C.2 Identify and describe relationships among inscribed angles, radii, and chords. G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Mathematical Practices 4 Model with mathematics. 1 Make sense of problems and persevere in solving them. Then/Now

You found measures of interior angles of polygons. (Lesson 6–1) Find measures of inscribed angles. Find measures of angles of inscribed polygons. Then/Now

inscribed angle intercepted arc Vocabulary

Concept

Concept

A. Find mX. Answer: mX = 43 Use Inscribed Angles to Find Measures Example 1

Use Inscribed Angles to Find Measures = 2(52) or 104 Example 1

A. Find mC. A. 47 B. 54 C. 94 D. 188 Example 1

B. A. 47 B. 64 C. 94 D. 96 Example 1

Concept

R  S R and S both intercept . Use Inscribed Angles to Find Measures ALGEBRA Find mR. R  S R and S both intercept . mR  mS Definition of congruent angles 12x – 13 = 9x + 2 Substitution x = 5 Simplify. Answer: So, mR = 12(5) – 13 or 47. Example 2

ALGEBRA Find mI. A. 4 B. 25 C. 41 D. 49 Example 2

Write a two-column proof. Given: Prove: ΔMNP  ΔLOP Use Inscribed Angles in Proofs Write a two-column proof. Given: Prove: ΔMNP  ΔLOP Proof: Statements Reasons 1. Given LO  MN 2. If minor arcs are congruent, then corresponding chords are congruent. Example 3

3. Definition of intercepted arc M intercepts and L intercepts . Use Inscribed Angles in Proofs Proof: Statements Reasons 3. Definition of intercepted arc M intercepts and L intercepts . M  L 4. Inscribed angles of the same arc are congruent. MPN  OPL 5. Vertical angles are congruent. ΔMNP  ΔLOP 6. AAS Congruence Theorem Example 3

Write a two-column proof. Given: Prove: ΔABE  ΔDCE Select the appropriate reason that goes in the blank to complete the proof below. Proof: Statements Reasons 1. Given AB  DC 2. If minor arcs are congruent, then corresponding chords are congruent. Example 3

3. Definition of intercepted arc D intercepts and A intercepts . Proof: Statements Reasons 3. Definition of intercepted arc D intercepts and A intercepts . D  A 4. Inscribed angles of the same arc are congruent. DEC  BEA 5. Vertical angles are congruent. ΔDCE  ΔABE 6. ____________________ Example 3

A. SSS Congruence Theorem B. AAS Congruence Theorem C. Definition of congruent triangles D. Definition of congruent arcs Example 3

Concept

ΔABC is a right triangle because C inscribes a semicircle. Find Angle Measures in Inscribed Triangles ALGEBRA Find mB. ΔABC is a right triangle because C inscribes a semicircle. mA + mB + mC = 180 Angle Sum Theorem (x + 4) + (8x – 4) + 90 = 180 Substitution 9x + 90 = 180 Simplify. 9x = 90 Subtract 90 from each side. x = 10 Divide each side by 9. Answer: So, mB = 8(10) – 4 or 76. Example 4

ALGEBRA Find mD. A. 8 B. 16 C. 22 D. 28 Example 4

Concept

Find Angle Measures INSIGNIAS An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find mS and mT. Example 5

Answer: So, mS = 90 and mT = 8(8) + 4 or 68. Find Angle Measures Since TSUV is inscribed in a circle, opposite angles are supplementary. mS + mV = 180 mU + mT = 180 mS + 90 = 180 (14x) + (8x + 4) = 180 mS = 90 22x + 4 = 180 22x = 176 x = 8 Answer: So, mS = 90 and mT = 8(8) + 4 or 68. Example 5

INSIGNIAS An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find mN. A. 48 B. 36 C. 32 D. 28 Example 5

End of the Lesson