25. A polygon is tangent to the circle. 26. If a central angle measures 25 , then the measure of its intercepted arc is 27. The part of a circle included between two of its points is a/an 28. If inscribed $$ \angle B $$ measures 48 , then the measure of its intercepted arc, $$ \overparen{A C} $$, is 29. If inscribed $$ \angle C $$ intercepts $$ \overparen{B D} $$, which measures 214 , then the angle measures 30. $$ \angle A $$ is formed by two tangents and measures 36 . The measure of intercepted $$ \overparen{B C} $$ is 144 . What is the measure of $$ \overparen{B D C} $$ ? 31. If central $$ \angle A $$ intercepts $$ \overparen{B C} $$, which measures 113 , then $$ \angle A $$ measures 32. Two chords intersect within a circle and form an angle measuring 65 . The measure of one intercepted arc is 79 . Let $$ \overparen{B C} $$ be the measure of the other arc. What is the measure of $$ \overparen{B C} $$ ?

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Sure, let's address each of your geometry questions (25-32) one by one:

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**25. A polygon is tangent to the circle.**

- **Answer:** Such a polygon is called a **tangential polygon** or a **circumscribed polygon**. In this case, each side of the polygon touches the circle at exactly one point, making the circle its incircle.

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**26. If a central angle measures 25°, then the measure of its intercepted arc is**

- **Answer:** The measure of the intercepted arc is equal to the measure of the central angle. Therefore, the intercepted arc measures **25°**.

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**27. The part of a circle included between two of its points is a/an**

- **Answer:** This part is called an **arc**. An arc is a continuous portion of the circumference between two points on the circle.

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**28. If inscribed $$ \angle B $$ measures 48°, then the measure of its intercepted arc, $$ \overparen{AC} $$, is**

- **Answer:** An inscribed angle is half the measure of its intercepted arc. Therefore:
  
  $$
  \text{Measure of } \overparen{AC} = 2 \times 48° = 96°
  $$

So, the intercepted arc $$ \overparen{AC} $$ measures **96°**.

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**29. If inscribed $$ \angle C $$ intercepts $$ \overparen{BD} $$, which measures 214°, then the angle measures**

- **Answer:** Again, an inscribed angle is half the measure of its intercepted arc. Therefore:
  
  $$
  \angle C = \frac{214°}{2} = 107°
  $$

So, $$ \angle C $$ measures **107°**.

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**30. $$ \angle A $$ is formed by two tangents and measures 36°. The measure of intercepted $$ \overparen{BC} $$ is 144°. What is the measure of $$ \overparen{BDC} $$?**

- **Answer:** When two tangents form an angle outside the circle, the measure of that angle is half the difference of the measures of the intercepted arcs. Given:

$$
  \angle A = \frac{|\overparen{BDC} - \overparen{BC}|}{2} = 36°
  $$
  
  We know $$ \overparen{BC} = 144° $$. Let $$ \overparen{BDC} = x $$:
  
  $$
  \frac{|x - 144°|}{2} = 36° \implies |x - 144°| = 72°
  $$
  
  Solving for $$ x $$:
  
  $$
  x - 144° = 72° \implies x = 216°
  $$
  
  **Hence, $$ \overparen{BDC} $$ measures 216°.**

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**31. If central $$ \angle A $$ intercepts $$ \overparen{BC} $$, which measures 113°, then $$ \angle A $$ measures**

- **Answer:** A central angle is equal in measure to its intercepted arc. Therefore:

$$
  \angle A = 113°
  $$

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**32. Two chords intersect within a circle and form an angle measuring 65°. The measure of one intercepted arc is 79°. Let $$ \overparen{BC} $$ be the measure of the other arc. What is the measure of $$ \overparen{BC} $$?**

- **Answer:** When two chords intersect inside a circle, the measure of the angle formed is half the sum of the measures of the intercepted arcs. Let $$ \overparen{BC} = x $$:
  
  $$
  65° = \frac{79° + x}{2} \implies 130° = 79° + x \implies x = 51°
  $$
  
  **Thus, $$ \overparen{BC} $$ measures 51°.**

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Feel free to ask if you need further clarification on any of these problems!
**25. A polygon tangent to a circle is called a **tangential polygon** or a **circumscribed polygon**.** **26. If a central angle measures 25°, its intercepted arc also measures **25°**.** **27. The part of a circle between two points is called an **arc**.** **28. If an inscribed angle $$ \angle B $$ measures 48°, its intercepted arc $$ \overparen{AC} $$ measures **96°**.** **29. If an inscribed angle $$ \angle C $$ intercepts arc $$ \overparen{BD} $$ measuring 214°, then $$ \angle C $$ measures **107°**.** **30. Given that $$ \angle A $$ formed by two tangents measures 36° and intercepts arc $$ \overparen{BC} $$ of 144°, the measure of arc $$ \overparen{BDC} $$ is **216°**.** **31. If central angle $$ \angle A $$ intercepts arc $$ \overparen{BC} $$ of 113°, then $$ \angle A $$ measures **113°**.** **32. If two chords intersect within a circle and form an angle of 65°, with one intercepted arc measuring 79°, the measure of the other arc $$ \overparen{BC} $$ is **51°**.**

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