The sum of the measures of the interior angles of a nonagon is 19809. Classify the polygon by the number of sides. 3. Find the value of x. 4. Find the value of x. 7z1 96 1019 1678 5. Find the measure of an interior and exterior angle of a regular pentagon. 6. Each interior angle of a regular n-gon has a measure of 99. Find the value of n.
The Correct Answer and Explanation is:
Here are the solutions to the provided geometry problems, followed by a detailed explanation of the methods used:
- Sum of interior angles of a nonagon: A nonagon has 9 sides. The sum of interior angles is calculated using the formula
(n−2)×180(n – 2) × 180
(9−2)×180=7×180=1260(9 – 2) × 180 = 7 × 180 = 1260
So, the correct sum is 1260 degrees, not 19809. The value 19809 appears to be a typographical error.
- Classify a polygon with a 1980-degree interior angle sum: Using the same formula:
(n−2)×180=1980(n – 2) × 180 = 1980
Divide both sides by 180:
n−2=11n – 2 = 11
n=13n = 13
The polygon is a 13-gon, also called a tridecagon.
- Find x when interior angles are 121°, 96°, 101°, 162°, and x: The polygon has 5 sides, so:
(5−2)×180=540(5 – 2) × 180 = 540
121+96+101+162+x=540121 + 96 + 101 + 162 + x = 540
x=540−(121+96+101+162)=540−480=60x = 540 – (121 + 96 + 101 + 162) = 540 – 480 = 60
So, x = 60 degrees.
- Find x from the equation: 45° + x + 2x + 77° = 360°: Combine like terms:
3x+122=3603x + 122 = 360
3x=2383x = 238
x=79.33x = 79.33
Thus, x ≈ 79.33 degrees.
- Interior and exterior angles of a regular pentagon: Interior angle =
(5−2)×180/5=108(5 – 2) × 180 / 5 = 108
Exterior angle =
180−108=72180 – 108 = 72
So, interior = 108°, exterior = 72°.
- Regular n-gon with interior angle of 99°: Use the formula:
(n−2)×180/n=99(n – 2) × 180 / n = 99
Multiply both sides by n:
180n−360=99n180n – 360 = 99n
81n=36081n = 360
n=360/81=4.44n = 360 / 81 = 4.44
Since n must be an integer, there is no regular polygon with each interior angle measuring exactly 99 degrees.
Explanation
To solve problems involving polygons, we rely on formulas derived from the relationships between sides, angles, and the geometric structure. A key formula is the sum of interior angles:
(n−2)×180(n – 2) × 180
This tells us that each additional side adds 180 degrees to the total interior sum. For regular polygons where all angles are equal, we divide that sum by the number of sides to find one interior angle. Exterior angles, on the other hand, always add up to 360 degrees for any polygon, so each individual exterior angle in a regular polygon is simply
360/n360 / n
.
In the case of a 13-gon, reversing the interior sum formula allowed us to solve for n. When we calculated the unknown angle x in problem 3, we summed all known angles and subtracted from the total expected sum. For problem 4, we used algebra to solve for a variable appearing in more than one term. The pentagon problem applied both formulas directly to determine standard angle values. For problem 6, we saw that an interior angle of 99 degrees does not correspond to a whole number of sides, which means such a regular polygon cannot exist in Euclidean geometry.
By consistently applying these foundational geometric rules, we can solve complex-looking problems through structured reasoning.
