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 \( \overparen{B C} \) ?

Sure, let's address each of your geometry questions (25-32) one by one: --- **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. --- **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°**. --- **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. --- **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°**. --- **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°**. --- **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°.** --- **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: \[ \angle A = 113° \] --- **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°.** --- Feel free to ask if you need further clarification on any of these problems!