in exercises 1–4, find the sum of the measures of the interior angles of the indicated convex polygon. (see example 1.)
1. nonagon 2. 14 - gon 3. 16 - gon 4. 20 - gon

in exercises 5–8, the sum of the measures of the interior angles of a convex polygon is given. classify the polygon by the number of sides. (see example 2.)
5. 720° 6. 1080° 7. 2520° 8. 3240°

in exercises 9–14, find the value of x. (see example 3.)
9. quadrilateral with angles 100°, 130°, 66°, x° 10. quadrilateral with angles 103°, 133°, 58°, x°
11. pentagon with angles x°, 154°, 88°, 29° 12. quadrilateral with angles x°, 92°, 101°, 68°
13. hexagon with angles 102°, 146°, 120°, 124°, 158°, x° 14. pentagon with angles 86°, 140°, 138°, 59°, x°

Question

in exercises 1–4, find the sum of the measures of the interior angles of the indicated convex polygon. (see example 1.)
1. nonagon  2. 14 - gon  3. 16 - gon  4. 20 - gon

in exercises 5–8, the sum of the measures of the interior angles of a convex polygon is given. classify the polygon by the number of sides. (see example 2.)
5. 720°  6. 1080°  7. 2520°  8. 3240°

in exercises 9–14, find the value of x. (see example 3.)
9. quadrilateral with angles 100°, 130°, 66°, x°  10. quadrilateral with angles 103°, 133°, 58°, x°
11. pentagon with angles x°, 154°, 88°, 29°  12. quadrilateral with angles x°, 92°, 101°, 68°
13. hexagon with angles 102°, 146°, 120°, 124°, 158°, x°  14. pentagon with angles 86°, 140°, 138°, 59°, x°

Accepted Answer

### Exercise 1: Nonagon
# Explanation:
## Step 1: Recall the formula for the sum of interior angles of a convex polygon.
The formula is $ S = (n - 2) \times 180^\circ $, where $ n $ is the number of sides. A nonagon has $ n = 9 $ sides.

## Step 2: Substitute $ n = 9 $ into the formula.
$ S = (9 - 2) \times 180^\circ = 7 \times 180^\circ $

## Step 3: Calculate the result.
$ 7 \times 180^\circ = 1260^\circ $

# Answer:
The sum of the interior angles of a nonagon is $ 1260^\circ $.

### Exercise 2: 14 - gon
# Explanation:
## Step 1: Use the formula $ S = (n - 2) \times 180^\circ $ with $ n = 14 $.

## Step 2: Substitute $ n = 14 $.
$ S = (14 - 2) \times 180^\circ = 12 \times 180^\circ $

## Step 3: Compute the product.
$ 12 \times 180^\circ = 2160^\circ $

# Answer:
The sum of the interior angles of a 14 - gon is $ 2160^\circ $.

### Exercise 3: 16 - gon
# Explanation:
## Step 1: Apply the formula $ S = (n - 2) \times 180^\circ $ with $ n = 16 $.

## Step 2: Substitute $ n = 16 $.
$ S = (16 - 2) \times 180^\circ = 14 \times 180^\circ $

## Step 3: Calculate.
$ 14 \times 180^\circ = 2520^\circ $

# Answer:
The sum of the interior angles of a 16 - gon is $ 2520^\circ $.

### Exercise 4: 20 - gon
# Explanation:
## Step 1: Use the formula $ S = (n - 2) \times 180^\circ $ with $ n = 20 $.

## Step 2: Substitute $ n = 20 $.
$ S = (20 - 2) \times 180^\circ = 18 \times 180^\circ $

## Step 3: Compute.
$ 18 \times 180^\circ = 3240^\circ $

# Answer:
The sum of the interior angles of a 20 - gon is $ 3240^\circ $.

### Exercise 5: Sum = $ 720^\circ $
# Explanation:
## Step 1: Use the formula $ S = (n - 2) \times 180^\circ $ and solve for $ n $.

## Step 2: Substitute $ S = 720^\circ $ into the formula.
$ 720^\circ = (n - 2) \times 180^\circ $

## Step 3: Divide both sides by $ 180^\circ $.
$ \frac{720^\circ}{180^\circ} = n - 2 $
$ 4 = n - 2 $

## Step 4: Solve for $ n $.
$ n = 4 + 2 = 6 $

# Answer:
The polygon has 6 sides (a hexagon).

### Exercise 6: Sum = $ 1080^\circ $
# Explanation:
## Step 1: Use $ S = (n - 2) \times 180^\circ $ and solve for $ n $.

## Step 2: Substitute $ S = 1080^\circ $.
$ 1080^\circ = (n - 2) \times 180^\circ $

## Step 3: Divide by $ 180^\circ $.
$ \frac{1080^\circ}{180^\circ} = n - 2 $
$ 6 = n - 2 $

## Step 4: Solve for $ n $.
$ n = 6 + 2 = 8 $

# Answer:
The polygon has 8 sides (an octagon).

### Exercise 7: Sum = $ 2520^\circ $
# Explanation:
## Step 1: Apply $ S = (n - 2) \times 180^\circ $ and solve for $ n $.

## Step 2: Substitute $ S = 2520^\circ $.
$ 2520^\circ = (n - 2) \times 180^\circ $

## Step 3: Divide by $ 180^\circ $.
$ \frac{2520^\circ}{180^\circ} = n - 2 $
$ 14 = n - 2 $

## Step 4: Solve for $ n $.
$ n = 14 + 2 = 16 $

# Answer:
The polygon has 16 sides (a 16 - gon).

### Exercise 8: Sum = $ 3240^\circ $
# Explanation:
## Step 1: Use $ S = (n - 2) \times 180^\circ $ and solve for $ n $.

## Step 2: Substitute $ S = 3240^\circ $.
$ 3240^\circ = (n - 2) \times 180^\circ $

## Step 3: Divide by $ 180^\circ $.
$ \frac{3240^\circ}{180^\circ} = n - 2 $
$ 18 = n - 2 $

## Step 4: Solve for $ n $.
$ n = 18 + 2 = 20 $

# Answer:
The polygon has 20 sides (a 20 - gon).

### Exercise 9: Quadrilateral (4 sides)
# Explanation:
## Step 1: Recall the sum of interior angles of a quadrilateral: $ (4 - 2) \times 180^\circ = 360^\circ $.

## Step 2: Let the angles be $ 100^\circ $, $ 130^\circ $, $ 66^\circ $, and $ x^\circ $. Their sum is $ 360^\circ $.

## Step 3: Set up the equation.
$ 100 + 130 + 66 + x = 360 $

## Step 4: Simplify the left - hand side.
$ 296 + x = 360 $

## Step 5: Solve for $ x $.
$ x = 360 - 296 = 64 $

# Answer:
$ x = 64^\circ $

### Exercise 10: Quadrilateral
# Explanation:
## Step 1: The sum of interior angles of a quadrilateral is $ 360^\circ $.

## Step 2: The angles are $ 103^\circ $, $ 133^\circ $, $ 58^\circ $, and $ x^\circ $.

## Step 3: Set up the equation.
$ 103 + 133 + 58 + x = 360 $

## Step 4: Simplify.
$ 294 + x = 360 $

## Step 5: Solve for $ x $.
$ x = 360 - 294 = 66 $

# Answer:
$ x = 66^\circ $

### Exercise 11: Quadrilateral? Wait, no, the figure has 4 angles? Wait, the figure shown has 4 angles? Wait, no, looking at the diagram, it seems to be a quadrilateral? Wait, no, the angles are $ x^\circ $, $ 154^\circ $, $ 88^\circ $, $ 29^\circ $? Wait, no, maybe it's a quadrilateral? Wait, the sum of interior angles of a quadrilateral is $ 360^\circ $.

# Explanation:
## Step 1: Sum of interior angles of a quadrilateral is $ 360^\circ $.

## Step 2: Angles are $ x $, $ 154^\circ $, $ 88^\circ $, $ 29^\circ $.

## Step 3: Equation: $ x + 154 + 88 + 29 = 360 $

## Step 4: Simplify: $ x + 271 = 360 $

## Step 5: Solve for $ x $: $ x = 360 - 271 = 89 $

# Answer:
$ x = 89^\circ $

### Exercise 12: Quadrilateral
# Explanation:
## Step 1: Sum of interior angles of a quadrilateral is $ 360^\circ $.

## Step 2: Angles are $ x $, $ 101^\circ $, $ 68^\circ $, $ 92^\circ $.

## Step 3: Equation: $ x + 101 + 68 + 92 = 360 $

## Step 4: Simplify: $ x + 261 = 360 $

## Step 5: Solve for $ x $: $ x = 360 - 261 = 99 $

# Answer:
$ x = 99^\circ $

### Exercise 13: Hexagon (6 sides)
# Explanation:
## Step 1: The sum of interior angles of a hexagon is $ (6 - 2) \times 180^\circ = 720^\circ $.

## Step 2: The angles are $ 102^\circ $, $ 146^\circ $, $ 120^\circ $, $ 124^\circ $, $ 158^\circ $, and $ x^\circ $.

## Step 3: Sum the known angles: $ 102 + 146 + 120 + 124 + 158 = 102+146 = 248; 248 + 120 = 368; 368+124 = 492; 492 + 158 = 650 $

## Step 4: Set up the equation: $ 650 + x = 720 $

## Step 5: Solve for $ x $: $ x = 720 - 650 = 70 $

# Answer:
$ x = 70^\circ $

### Exercise 14: Pentagon (5 sides)
# Explanation:
## Step 1: The sum of interior angles of a pentagon is $ (5 - 2) \times 180^\circ = 540^\circ $.

## Step 2: The angles are $ 86^\circ $, $ 140^\circ $, $ 138^\circ $, $ 59^\circ $, and $ x^\circ $.

## Step 3: Sum the known angles: $ 86+140 = 226; 226 + 138 = 364; 364+59 = 423 $

## Step 4: Set up the equation: $ 423 + x = 540 $

## Step 5: Solve for $ x $: $ x = 540 - 423 = 117 $

# Answer:
$ x = 117^\circ $

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QUESTION IMAGE

Question

Explanation:

Exercise 1: Nonagon

Step 1: Recall the formula for the sum of interior angles of a convex polygon.

Step 2: Substitute \( n = 9 \) into the formula.

Step 3: Calculate the result.

Step 1: Use the formula \( S = (n - 2) \times 180^\circ \) with \( n = 14 \).

Step 2: Substitute \( n = 14 \).

Step 3: Compute the product.

Step 1: Apply the formula \( S = (n - 2) \times 180^\circ \) with \( n = 16 \).

Step 2: Substitute \( n = 16 \).

Step 3: Calculate.

Answer:

Exercise 2: 14 - gon