Find the number of sides a regular polygon must have to meet each condition. 32. Each interior angle measure equals each exterior angle measure. 33. Each interior angle measure is four times the measure of each exterior angle. 34. Each exterior angle measure is one eighth the measure of each interior angle. Name the convex polygon whose interior angle measures have each given sum. \( \begin{array}{llll}\text { 35. } 540^{\circ} & \text { 36. } 900^{\circ} & \text { 37. } 1800^{\circ} & \text { 38. } 2520^{\circ}\end{array} \)

Let's solve each problem step by step. --- ### **Problem 32: Each interior angle measure equals each exterior angle measure.** **Solution:** - Let each exterior angle be \( x \) degrees. - Since interior and exterior angles are supplementary: \[ x + x = 180^\circ \implies 2x = 180^\circ \implies x = 90^\circ \] - The sum of all exterior angles of a polygon is \( 360^\circ \): \[ n = \frac{360^\circ}{x} = \frac{360^\circ}{90^\circ} = 4 \] **Answer:** The polygon has **4 sides** (a **quadrilateral**). --- ### **Problem 33: Each interior angle measure is four times the measure of each exterior angle.** **Solution:** - Let each exterior angle be \( x \) degrees. - Then, each interior angle is \( 4x \) degrees. - Since they are supplementary: \[ x + 4x = 180^\circ \implies 5x = 180^\circ \implies x = 36^\circ \] - Number of sides: \[ n = \frac{360^\circ}{x} = \frac{360^\circ}{36^\circ} = 10 \] **Answer:** The polygon has **10 sides** (a **decagon**). --- ### **Problem 34: Each exterior angle measure is one eighth the measure of each interior angle.** **Solution:** - Let each exterior angle be \( x \) degrees. - Then, each interior angle is \( 8x \) degrees. - Since they are supplementary: \[ x + 8x = 180^\circ \implies 9x = 180^\circ \implies x = 20^\circ \] - Number of sides: \[ n = \frac{360^\circ}{x} = \frac{360^\circ}{20^\circ} = 18 \] **Answer:** The polygon has **18 sides** (an **octadecagon**). --- ### **Problem 35: Name the convex polygon whose interior angle measures sum to \( 540^\circ \).** **Solution:** - Sum of interior angles: \[ 180(n - 2) = 540^\circ \implies n - 2 = 3 \implies n = 5 \] - A 5-sided polygon is called a **pentagon**. **Answer:** **Pentagon**. --- ### **Problem 36: Name the convex polygon whose interior angle measures sum to \( 900^\circ \).** **Solution:** - Sum of interior angles: \[ 180(n - 2) = 900^\circ \implies n - 2 = 5 \implies n = 7 \] - A 7-sided polygon is called a **heptagon**. **Answer:** **Heptagon**. --- ### **Problem 37: Name the convex polygon whose interior angle measures sum to \( 1800^\circ \).** **Solution:** - Sum of interior angles: \[ 180(n - 2) = 1800^\circ \implies n - 2 = 10 \implies n = 12 \] - A 12-sided polygon is called a **dodecagon**. **Answer:** **Dodecagon**. --- ### **Problem 38: Name the convex polygon whose interior angle measures sum to \( 2520^\circ \).** **Solution:** - Sum of interior angles: \[ 180(n - 2) = 2520^\circ \implies n - 2 = 14 \implies n = 16 \] - A 16-sided polygon is called a **hexadecagon**. **Answer:** **Hexadecagon**. --- ### **Summary of Answers:** - **32.** 4 sides (quadrilateral) - **33.** 10 sides (decagon) - **34.** 18 sides (octadecagon) - **35.** Pentagon - **36.** Heptagon - **37.** Dodecagon - **38.** Hexadecagon