Question

name martina cambre pacheco date: 12-01-2025 period: 4
1d. notes: polygons - interior angles
essential question: how can we find interior/exterior angles and sides of a polygon?
1.) what do we know about the sum of the interior angles of a triangle? explain how we know this.
180, if you are too separate the angles
draw all the possible diagonals from one vertex of the polygon. then complete the table below.
2.) triangle abc image 3.) quadrilateral abcd image 4.) pentagon abcde image 5.) hexagon image
number of sides of the polygon n= 3 4 5 6 7 8 9 16
number of diagonals drawn n-3 0 1 2 3 4 5 6 13
number of triangles formed n-2 1 -1 0 1 5 6 4 11
sum of the measure of the interior angles 180(n-2) 180 360 540 720 900 1080 1260 1620
6.) what do you notice about the sum of the measures of the interior angles of a polygon as the number of sides of each polygon increases by 1?
7.) if a polygon has n sides, how many triangles are formed by drawing all diagonals from one vertex?
8.) if a polygon has n sides, how many diagonals can you create from one vertex?
9.) what is the sum of the measures of the interior angles of an n - sided polygon?
use the formulas we created to solve the following:
10.) how many diagonals can you create from one vertex of an 87 sided polygon? n - 3
87 - 3 = 84 diagonals
11.) how many triangles are formed from a 249 sided polygon? n - 2
249 - 2 = 247 triangles
12.) what is the sum of the measures of the interior angles of a polygon with 109 sides? 180(n - 2)
180(109 - 2)
180(107)
= 19260°
13.) what is the sum of the measures of the interior angles of a decagon?
180(10 - 2)
180 * 8 = 1440°

Explanation:

Answer:

The sum of the interior angles of a triangle is \(180^{\circ}\). This can be verified by cutting out the angles of a triangle and arranging them to form a straight line (a \(180^{\circ}\) angle).
For a polygon with \(n\) sides:

• The number of triangles formed by drawing diagonals from one vertex is \(n - 2\).
• The number of diagonals drawn from one vertex is \(n - 3\).
• The sum of the interior angles is \(180(n - 2)\) degrees.

1. \(n - 2\)
2. \(n - 3\)
3. \(180(n - 2)\)
4. \(87-3 = 84\)
5. \(249 - 2=247\)
6. \(180\times(109 - 2)=19260^{\circ}\)
7. A decagon has \(n = 10\) sides. \(180\times(10 - 2)=1440^{\circ}\)