example 1: find the sum of the measures of the interior angles of the indicated convex polygon.
a) nonagon
b) dodecagon
c) 23 - gon

example 2: the sum of the measures of the interior angles of a convex polygon is given. classify the polygon by the number of sides.
a) 1440°
b) 2700°
c) 8820°

example 3: find the value of x.
a) diagram with 93°, 75°, x°, 2x°
b) diagram with 7x°, 3x°, 20°, 6x°, 100°
c) diagram with 2x°, a right angle, 60°, 4x°, x°
d) diagram with 24x°, 10x°, 8x°, 30x°

Question

example 1: find the sum of the measures of the interior angles of the indicated convex polygon.
a) nonagon
b) dodecagon
c) 23 - gon

example 2: the sum of the measures of the interior angles of a convex polygon is given. classify the polygon by the number of sides.
a) 1440°
b) 2700°
c) 8820°

example 3: find the value of x.
a) diagram with 93°, 75°, x°, 2x°
b) diagram with 7x°, 3x°, 20°, 6x°, 100°
c) diagram with 2x°, a right angle, 60°, 4x°, x°
d) diagram with 24x°, 10x°, 8x°, 30x°

Accepted Answer

### Example 1
#### a) Nonagon (9 - sided polygon)
# Explanation:
## Step 1: Recall the formula for the sum of interior angles of a convex polygon, which is $ S=(n - 2)\times180^{\circ} $, where $ n $ is the number of sides.
For a nonagon, $ n = 9 $.
## Step 2: Substitute $ n = 9 $ into the formula.
$ S=(9 - 2)\times180^{\circ}=7\times180^{\circ}=1260^{\circ} $
# Answer: $ 1260^{\circ} $

#### b) Dodecagon (12 - sided polygon)
# Explanation:
## Step 1: Use the formula $ S=(n - 2)\times180^{\circ} $, with $ n = 12 $.
## Step 2: Substitute $ n = 12 $ into the formula.
$ S=(12 - 2)\times180^{\circ}=10\times180^{\circ}=1800^{\circ} $
# Answer: $ 1800^{\circ} $

#### c) 23 - gon (23 - sided polygon)
# Explanation:
## Step 1: Apply the formula $ S=(n - 2)\times180^{\circ} $, where $ n = 23 $.
## Step 2: Substitute $ n = 23 $ into the formula.
$ S=(23 - 2)\times180^{\circ}=21\times180^{\circ}=3780^{\circ} $
# Answer: $ 3780^{\circ} $

### Example 2
The formula for the sum of interior angles of a convex polygon is $ S=(n - 2)\times180^{\circ} $, and we can solve for $ n $ as $ n=\frac{S}{180^{\circ}}+2 $.

#### a) $ S = 1440^{\circ} $
# Explanation:
## Step 1: Use the formula $ n=\frac{S}{180^{\circ}}+2 $. Substitute $ S = 1440^{\circ} $.
$ n=\frac{1440^{\circ}}{180^{\circ}}+2 $
## Step 2: Calculate $ \frac{1440^{\circ}}{180^{\circ}}=8 $, then $ n = 8 + 2=10 $.
So, it is a decagon (10 - sided polygon).
# Answer: Decagon (10 - sided polygon)

#### b) $ S = 2700^{\circ} $
# Explanation:
## Step 1: Apply the formula $ n=\frac{S}{180^{\circ}}+2 $. Substitute $ S = 2700^{\circ} $.
$ n=\frac{2700^{\circ}}{180^{\circ}}+2 $
## Step 2: Calculate $ \frac{2700^{\circ}}{180^{\circ}} = 15 $, then $ n=15 + 2 = 17 $.
So, it is a 17 - gon.
# Answer: 17 - gon

#### c) $ S = 8820^{\circ} $
# Explanation:
## Step 1: Use the formula $ n=\frac{S}{180^{\circ}}+2 $. Substitute $ S = 8820^{\circ} $.
$ n=\frac{8820^{\circ}}{180^{\circ}}+2 $
## Step 2: Calculate $ \frac{8820^{\circ}}{180^{\circ}}=49 $, then $ n = 49+2 = 51 $.
So, it is a 51 - gon.
# Answer: 51 - gon

### Example 3
The sum of the exterior angles of any convex polygon is $ 360^{\circ} $ (for triangles and polygons, the sum of exterior angles is always $ 360^{\circ} $ when considering one exterior angle at each vertex).

#### a) Angles: $ 93^{\circ},75^{\circ},x^{\circ},2x^{\circ} $
# Explanation:
## Step 1: The sum of exterior angles is $ 360^{\circ} $, so $ 93 + 75+x + 2x=360 $.
## Step 2: Combine like terms: $ 168+3x = 360 $.
## Step 3: Subtract 168 from both sides: $ 3x=360 - 168 = 192 $.
## Step 4: Divide both sides by 3: $ x=\frac{192}{3}=64 $.
# Answer: $ x = 64 $

#### b) Angles: $ 7x^{\circ},3x^{\circ},20^{\circ},6x^{\circ},100^{\circ} $
# Explanation:
## Step 1: Sum of exterior angles is $ 360^{\circ} $, so $ 7x+3x + 20+6x+100 = 360 $.
## Step 2: Combine like terms: $ 16x+120 = 360 $.
## Step 3: Subtract 120 from both sides: $ 16x=360 - 120 = 240 $.
## Step 4: Divide both sides by 16: $ x=\frac{240}{16}=15 $.
# Answer: $ x = 15 $

#### c) Angles: $ 90^{\circ} $ (right angle), $ 60^{\circ},2x^{\circ},4x^{\circ},x^{\circ} $
# Explanation:
## Step 1: Sum of exterior angles is $ 360^{\circ} $, so $ 90+60 + 2x+4x+x=360 $.
## Step 2: Combine like terms: $ 150 + 7x=360 $.
## Step 3: Subtract 150 from both sides: $ 7x=360 - 150 = 210 $.
## Step 4: Divide both sides by 7: $ x=\frac{210}{7}=30 $.
# Answer: $ x = 30 $

#### d) Angles: $ 24x^{\circ},10x^{\circ},8x^{\circ},30x^{\circ} $
# Explanation:
## Step 1: Sum of exterior angles is $ 360^{\circ} $, so $ 24x+10x + 8x+30x=360 $.
## Step 2: Combine like terms: $ 72x=360 $.
## Step 3: Divide both sides by 72: $ x=\frac{360}{72}=5 $.
# Answer: $ x = 5 $

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QUESTION IMAGE

Question

Explanation:

Example 1

a) Nonagon (9 - sided polygon)

Step 1: Recall the formula for the sum of interior angles of a convex polygon, which is \( S=(n - 2)\times180^{\circ} \), where \( n \) is the number of sides.

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

Step 1: Use the formula \( S=(n - 2)\times180^{\circ} \), with \( n = 12 \).

Step 2: Substitute \( n = 12 \) into the formula.

Step 1: Apply the formula \( S=(n - 2)\times180^{\circ} \), where \( n = 23 \).

Step 2: Substitute \( n = 23 \) into the formula.

Answer:

b) Dodecagon (12 - sided polygon)