Investigation: What is the interior angle sum of any triangle?
Enter only a number.
Investigation: What is the interior angle sum of any quadrilateral?
Enter only a number.
Investigation: What is the interior angle sum of any pentagon?
Enter only a number.
Extension: What pattern do you notice as you find the interior angle sums of these first 3 polygons? Do you think this pattern applies to all other polygons as well? Explain.
Exploration:
Manipulate the applet to create a polygon with 4 sides.
Drag the triangles slider to the right.
What is the minimum number of triangles that can be combined to form any quadrilateral?
Enter only a number.
Exploration: Multiply the number of triangles it takes to form a quadrilateral by the interior angle sum of a triangle. What is the product? Enter only a number.
Comparison: Is this the same interior angle sum you found for quadrilaterals in item 2?
If not, consider checking your work.
Exploration:
Manipulate the applet to create a polygon with 5 sides.
Drag the triangles slider to the far right.
What is the minimum number of triangles that can be combined to form any pentagon?
Enter only a number.
Exploration: Multiply the number of triangles it takes to form a pentagon by the interior angle sum of a triangle. What is the product? Enter only a number.
Comparison: Is this the same interior angle sum you found for pentagons in item 3?
If not, consider checking your work.
Investigation: Use inductive reasoning and intuition to complete the table.
Investigation: How does the number of sides of a polygon relate to the minimum number of triangles that can be combined to form the polygon?
Theorem: The relationship between the number of sides of a polygon and its interior angle sum integrates both the minimum number of triangles that combine to form the polygon and the interior angle sum of triangles into a single theorem.
This theorem is called the Polygon Angle Sum Theorem.
Let S = interior angle sum of a polygon and n = the number of sides of the polygon.
Given what you know, which of the formulas below represents the Polygon Angle Sum Theorem?
Do You Understand? Why is the word minimum important when naming the minimum number of triangles that compose polygons?