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Angles Inside the Triangle

Explore circle theorems to find angles formed by tangents, chords, and secants inside and outside circles. Practice constructing viable arguments with real-world examples.

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Angles Inside the Triangle

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  1. Angles Inside the Triangle Concept 55a

  2. Lesson Menu Five-Minute Check (over Lesson 10–5) CCSS Then/Now New Vocabulary Theorem 10.12 Example 1: Use Intersecting Chords or Secants Theorem 10.13 Example 2: Use Intersecting Secants and Tangents Theorem 10.14 Example 3: Use Tangents and Secants that Intersect Outside a Circle Example 4: Real-World Example: Apply Properties of Intersecting Secants Concept Summary: Circle and Angle Relationships

  3. CCSS Content Standards Reinforcement of G.C.4 Construct a tangent line from a point outside a given circle to the circle. Mathematical Practices 3 Construct viable arguments and critique the reasoning of others. 1 Make sense of problems and persevere in solving them.

  4. Then/Now You found measures of segments formed by tangents to a circle. • Find measures of angles formed by lines intersecting on or inside a circle. • Find measures of angles formed by lines intersecting outside the circle.

  5. Vocabulary • Tangent – a line or segment that intersects a circle at exactly one point. • Point of tangency – the point were a tangent and circle meet. • common tangent – a line, ray, or segment that is tangent to two circles in the same plane.

  6. 1. Copy the figure and draw the common tangents. If no common tangent exists, state no common tangent. Answer:These circles have no common tangents. Any tangent of the inner circle will intercept the outer circle in two points.

  7. 2. Copy the figure and draw the common tangents. If no common tangent exists, state no common tangent. Answer:These circles have 2 common tangents.

  8. 3. Copy the figure and draw the common tangents to determine how many there are. If no common tangent exists, choose no common tangent. 4 common tangents

  9. 4. Copy the figure and draw the common tangents to determine how many there are. If no common tangent exists, choose no common tangent. 3 common tangents

  10. Vocabulary • Secant – a line that intersects a circle in exactly two points.

  11. Concept

  12. 5. Find x. Answer:x = 82

  13. 6. Find x. y Step 1Find mVZW. mWZX = 180 – mVZW x = 180 – 79 x = 101 Answer:x = 101 Step 2Find mWZX.

  14. 7. Find x. A. 92 B. 95 C. 98 D. 104

  15. 8. Find x. A. 92 B. 95 C. 97 D. 102

  16. 9. Find x.

  17. 10. Find x. A. 96 B. 99 C. 101 D. 104

  18. Angles On the Triangle Concept 55B

  19. 11. Find mQPS. Answer:mQPS= 125

  20. 12. Answer:

  21. 13. Find mFGI. A. 98 B. 108 C. 112.5 D. 118.5

  22. 14. A. 99 B. 148.5 C. 162 D. 198

  23. Angles Outside the Triangle Concept 55C

  24. Concept

  25. 15. Find m  R.

  26. 16.

  27. 17.

  28. 18. A. 23 B. 26 C. 29 D. 32

  29. 19. A. 194 B. 202 C. 210 D. 230 x 360 – x

  30. 20.

  31. 21. A. 25 B. 35 C. 40 D. 45

  32. Concept

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