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Lesson Menu 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

Over Lesson 10–3 5-Minute Check 1 A.60 B.70 C.80 D.90

Over Lesson 10–3 5-Minute Check 2 A.40 B.45 C.50 D.55

Over Lesson 10–3 5-Minute Check 3 A.40 B.45 C.50 D.55

Over Lesson 10–3 5-Minute Check 4 A.40 B.30 C.25 D.22.5

Over Lesson 10–3 5-Minute Check 5 A.24.6 B.26.8 C.28.4 D.30.2

Over Lesson 10–3 5-Minute Check 6 A. B. C. D.

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.

Vocabulary inscribed angle intercepted arc

Concept

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

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

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

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

Concept

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

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

Example 3 Use Inscribed Angles in Proofs Write a two-column proof. Given: Prove: ΔMNP  ΔLOP 1. Given Proof: StatementsReasons LO  MN2. If minor arcs are congruent, then corresponding chords are congruent.

Example 3 Use Inscribed Angles in Proofs Proof: StatementsReasons  M   L 4. Inscribed angles of the same arc are congruent.  MPN   OPL5. Vertical angles are congruent. ΔMNP  ΔLOP6. AAS Congruence Theorem 3. Definition of intercepted arc  M intercepts and  L intercepts.

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. 1. Given Proof: StatementsReasons AB  DC2. If minor arcs are congruent, then corresponding chords are congruent.

Example 3 Proof: StatementsReasons  D   A 4.Inscribed angles of the same arc are congruent.  DEC   BEA5.Vertical angles are congruent. ΔDCE  ΔABE6. ____________________ 3. Definition of intercepted arc  D intercepts and  A intercepts.

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

Concept

Example 4 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 = 180Substitution 9x + 90= 180Simplify. 9x= 90Subtract 90 from each side. x= 10Divide each side by 9. Answer: So, m  B = 8(10) – 4 or 76.

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

Concept

Example 5 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 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 = 9022x + 4= x= 176 x= 8 Answer: So, m  S = 90 and m  T = 8(8) + 4 or 68.

Example 5 A.48 B.36 C.32 D.28 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.

End of the Lesson