8.3 Independent Practice Page 221 Answer Key

Mastering Angles in Polygons: A Complete Guide to 8.3 Independent Practice Problems

Finding a reliable answer key for textbook practice can feel like a shortcut, but true mastery comes from understanding the why behind every solution. The problems on page 221 of a typical 8.3 section—often focused on angles in polygons—are designed to build a critical geometric skill: applying formulas to solve real-world shape problems. This guide doesn’t just list answers; it deconstructs the concepts, walks through representative problem types you’re likely to encounter, and equips you with the reasoning to solve any similar question confidently. Whether you’re using Go Math!, Big Ideas Math, or another standard curriculum, the principles here are universal.

The Core Concepts: Interior and Exterior Angles

Before tackling practice problems, solidify these foundational ideas. A polygon is a closed figure with straight sides. The key formulas are your tools:

• Sum of Interior Angles: (n - 2) × 180°, where n is the number of sides.
• Measure of One Interior Angle (Regular Polygon): [(n - 2) × 180°] / n.
• Sum of Exterior Angles (any convex polygon): Always 360°.
• Measure of One Exterior Angle (Regular Polygon): 360° / n.

A crucial relationship: at any vertex, the interior angle and its adjacent exterior angle are supplementary (sum to 180°). Confusing these is the most common error.

Deconstructing Common Problem Types from Page 221

While the exact problems vary by textbook, independent practice pages typically cluster questions into these categories. Let’s solve representative examples for each.

Problem Type 1: Finding the Sum of Interior Angles

Example: “What is the sum of the interior angles of a 15-gon?”

• Step 1: Identify n. Here, n = 15.
• Step 2: Apply the formula: (15 - 2) × 180° = 13 × 180°.
• Step 3: Calculate: 13 × 180 = 2,340.
• Answer: The sum is 2,340°.

Why it works: You’re essentially dividing the polygon into (n-2) triangles. Each triangle sums to 180°.

Problem Type 2: Finding One Interior Angle of a Regular Polygon

Example: “Find the measure of one interior angle of a regular decagon.”

• Step 1: A decagon has n = 10 sides.
• Step 2: First, find the sum: (10 - 2) × 180° = 8 × 180° = 1,440°.
• Step 3: Since it’s regular, all angles are equal. Divide the sum by n: 1,440° / 10 = 144°.
• Answer: Each interior angle measures 144°.

Alternate Method: You can combine the steps into one formula: [(n - 2) × 180°] / n.

Problem Type 3: Finding the Number of Sides Given an Interior Angle

Example: “The measure of one interior angle of a regular polygon is 150°. How many sides does it have?” This is a two-step algebra problem.

• Step 1: Set up the equation using the interior angle formula: [(n - 2) × 180°] / n = 150.
• Step 2: Solve for n:
  • Multiply both sides by n: (n - 2) × 180 = 150n.
  • Distribute: 180n - 360 = 150n.
  • Subtract 150n from both sides: 30n - 360 = 0.
  • Add 360: 30n = 360.
  • Divide by 30: n = 12.
• Answer: The polygon has 12 sides (a dodecagon).

Check: `(12-2)×180 = 1,800°; 1,800°/12 = 150°. Correct.

Problem Type 4: Working with Exterior Angles

Example: “One exterior angle of a regular polygon is 24°. How many sides does the polygon have?”

• Step 1: Recall the exterior angle formula for a regular polygon: 360° / n = exterior angle.
• Step 2: Set up the equation: 360 / n = 24.
• Step 3: Solve for n: n = 360 / 24 = 15.
• Answer: The polygon has 15 sides.

Example: “Find the sum of the exterior angles of a convex 22-gon.”

• Answer: This is the golden rule: 360°. It never changes for any convex polygon, regardless of the number of sides.

Problem Type 5: Finding an Exterior Angle Given the Interior Angle

Example: “An interior angle of a regular polygon is 165°. What is the measure of one exterior angle?”

• Step 1: Remember they are supplementary at a vertex.
• Step 2: Exterior Angle = 180° - Interior Angle.
• Step 3: 180° - 165° = 15°.
• Answer: Each exterior angle is 15°.
• Bonus: You could then find sides: 360° / 15° = 24 sides.

Problem Type 6: Solving for an Unknown Angle in a Polygon Diagram

These problems present a polygon with some angles given and one unknown (often labeled `x