Unit 8 Homework 1 Angles Of Polygons Answer Key

Introduction

Understanding the relationship between the number of sides of a polygon and its interior and exterior angles is a cornerstone of middle‑school geometry. Unit 8, Homework 1 – Angles of Polygons challenges students to apply formulas, draw diagrams, and solve word problems that reinforce this concept. This article walks through the key ideas, provides step‑by‑step solutions for each problem, and presents a complete answer key that teachers and learners can use for self‑assessment. By the end of the guide, readers will not only have the correct answers but also a solid grasp of why those answers are true, making future geometry tasks far less intimidating That's the part that actually makes a difference. Simple as that..

1. Core Concepts Reviewed in Unit 8

1.1 Interior Angle Formula

For any convex polygon with n sides, the sum of the interior angles is

[ \text{Sum of interior angles}= (n-2)\times 180^\circ . ]

If the polygon is regular (all sides and angles equal), each interior angle equals

[ \text{Each interior angle}= \frac{(n-2)\times 180^\circ}{n}. ]

1.2 Exterior Angle Formula

The exterior angle at each vertex of a regular polygon measures

[ \text{Each exterior angle}= \frac{360^\circ}{n}. ]

Because interior and exterior angles are supplementary,

[ \text{Interior angle}+ \text{Exterior angle}=180^\circ . ]

1.3 Types of Polygons

• Triangle (3‑sided) – simplest polygon, interior sum = 180°.
• Quadrilateral (4‑sided) – interior sum = 360°.
• Pentagon (5‑sided) – interior sum = 540°.
• Hexagon (6‑sided) – interior sum = 720°.

These values are often the first numbers students encounter on the homework sheet.

2. How to Approach the Homework Problems

2.1 Identify What Is Asked

Read each question carefully. Is it asking for:

• The sum of interior angles?
• The measure of a single interior or exterior angle of a regular polygon?
• The number of sides given an angle measure?

2.2 Choose the Correct Formula

• Sum of interior angles → use ((n-2) \times 180^\circ).
• Single interior angle of a regular polygon → divide the sum by n.
• Single exterior angle → use (360^\circ / n).

2.3 Solve Algebraically When Needed

If the problem provides an angle and asks for n, set up an equation and solve for n. Remember that n must be a whole number greater than 2.

2.4 Double‑Check with Reasoning

After obtaining a numeric answer, verify it makes sense:

• Does the angle fall within the possible range for a polygon?
• Does the calculated n produce an integer interior/exterior angle?

3. Step‑by‑Step Solutions for Each Question

Below is a typical set of ten questions that appear on Unit 8 Homework 1. The solutions follow the systematic approach described above.

Question 1

Find the sum of the interior angles of a heptagon (7‑sided polygon).

Solution
[ \text{Sum}= (7-2)\times180^\circ =5\times180^\circ =900^\circ . ]

Answer: 900°.

Question 2

A regular octagon has each interior angle equal to ___ degrees.

Solution
[ \text{Each interior}= \frac{(8-2)\times180^\circ}{8}= \frac{6\times180^\circ}{8}= \frac{1080^\circ}{8}=135^\circ . ]

Answer: 135° Easy to understand, harder to ignore..

Question 3

What is the measure of each exterior angle of a regular nonagon (9‑sided polygon)?

Solution
[ \text{Each exterior}= \frac{360^\circ}{9}=40^\circ . ]

Answer: 40°.

Question 4

If each interior angle of a regular polygon measures 150°, how many sides does the polygon have?

Solution
Set up the equation

[ \frac{(n-2)\times180^\circ}{n}=150^\circ . ]

Multiply both sides by n:

[ (n-2)\times180 =150n . ]

Expand and simplify:

[ 180n-360 =150n \ 30n =360 \ n =12 . ]

Answer: 12 sides (a regular dodecagon) Simple as that..

Question 5

Find the sum of the interior angles of a polygon with 12 sides.

Solution
[ \text{Sum}= (12-2)\times180^\circ =10\times180^\circ =1800^\circ . ]

Answer: 1800° It's one of those things that adds up..

Question 6

A regular polygon has each exterior angle of 24°. Determine the number of sides.

Solution
[ n = \frac{360^\circ}{24^\circ}=15 . ]

Answer: 15 sides No workaround needed..

Question 7

What is the measure of each interior angle of a regular decagon (10‑sided polygon)?

Solution
[ \text{Each interior}= \frac{(10-2)\times180^\circ}{10}= \frac{8\times180^\circ}{10}= \frac{1440^\circ}{10}=144^\circ . ]

Answer: 144° Turns out it matters..

Question 8

The sum of the interior angles of a polygon is 1260°. How many sides does the polygon have?

Solution
[ (n-2)\times180^\circ =1260^\circ \ 180n -360 =1260 \ 180n =1620 \ n =9 . ]

Answer: 9 sides (a nonagon) The details matter here..

Question 9

If a regular polygon has an interior angle of 108°, identify the polygon.

Solution
Set up:

[ \frac{(n-2)\times180^\circ}{n}=108^\circ . ]

Multiply:

[ 180n-360 =108n \ 72n =360 \ n =5 . ]

Answer: A regular pentagon.

Question 10

A student claims that a regular polygon with 7 sides must have each exterior angle equal to 60°. Is the claim correct? Explain.

Solution
Calculate the true exterior angle:

[ \frac{360^\circ}{7}\approx 51.43^\circ . ]

Since 60° ≠ 51.43°, the claim is incorrect. That said, the correct exterior angle is about 51. 43°, and the corresponding interior angle is 180° − 51.43° ≈ 128.57°.

Answer: No; the correct exterior angle is approximately 51.43° Easy to understand, harder to ignore..

4. Complete Answer Key

5. Frequently Asked Questions

Q1: Why does the sum of interior angles increase by 180° each time I add a side?

A: Adding a side to a polygon is equivalent to attaching an extra triangle to the shape. Since a triangle’s interior angles sum to 180°, each new side contributes another 180° to the total.

Q2: Can a regular polygon have an interior angle of 179°?

A: Yes, but it would need a very large number of sides. Solving (\frac{(n-2)180}{n}=179) gives (n≈360). So a regular 360‑gon has interior angles of 179°.

Q3: Do the formulas work for concave polygons?

A: The sum of interior angles formula ((n-2)180^\circ) holds for any simple polygon, convex or concave. On the flip side, the regular polygon formulas for single interior or exterior angles only apply to convex, equiangular shapes.

Q4: How can I quickly check my answer for the number of sides?

A: Verify that the computed n yields an integer angle when you plug it back into the interior or exterior formula. If you get a fraction, the original angle value was likely rounded or the polygon is not regular.

Q5: What is the relationship between interior and exterior angles in a regular polygon?

A: They are supplementary: interior + exterior = 180°. Because of this, the interior angle can also be found by (180^\circ - \frac{360^\circ}{n}).

6. Tips for Mastering Polygon Angle Problems

1. Memorize the three anchor values – triangles (180°), quadrilaterals (360°), pentagons (540°). They serve as quick reference points.
2. Write the formula first before substituting numbers; this reduces careless errors.
3. Check units – always keep the degree symbol (°) to avoid mixing with radian measures.
4. Use a calculator for division only when necessary; many angles resolve to whole numbers, which signals a correct integer n.
5. Draw a rough sketch of the polygon when the problem involves visual reasoning (e.g., “find the missing angle in a diagram”). Visual cues often reveal whether the shape is regular or irregular.

7. Conclusion

The Unit 8 Homework 1 – Angles of Polygons assignment reinforces fundamental geometry skills that students will use throughout secondary math and beyond. Also, by mastering the interior‑angle sum formula, the regular‑polygon angle formulas, and the algebraic steps needed to solve for the number of sides, learners gain confidence in tackling more complex geometric reasoning. The answer key provided here serves as a reliable checkpoint, while the explanatory steps see to it that students understand why each answer is correct, not just what the answer is. Consistent practice with these concepts will turn the once‑daunting world of polygons into a familiar toolkit for any geometry challenge Which is the point..