To solve the problem of finding the sum of the interior angles of each polygon, we need to use a fundamental formula in geometry. The formula for the sum of the interior angles of a polygon with \( n \) sides is:
\[
\text{Sum of interior angles} = (n - 2) \times 180^\circ
\]
Here's a step-by-step explanation and solution for each polygon:
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Step 1: Understand the Formula
The formula \((n - 2) \times 180^\circ\) works because:
- Any polygon can be divided into \( n - 2 \) triangles.
- Each triangle has an interior angle sum of \( 180^\circ \).
- Therefore, the total sum of the interior angles of the polygon is \( (n - 2) \times 180^\circ \).
---
Step 2: Count the Sides of Each Polygon
For each polygon in the image, count the number of sides (\( n \)).
1.
First Polygon: This is a quadrilateral (4 sides).
2.
Second Polygon: This is a pentagon (5 sides).
3.
Third Polygon: This is a hexagon (6 sides).
4.
Fourth Polygon: This is a heptagon (7 sides).
5.
Fifth Polygon: This is an octagon (8 sides).
6.
Sixth Polygon: This is a nonagon (9 sides).
7.
Seventh Polygon: This is a decagon (10 sides).
---
Step 3: Apply the Formula
Now, use the formula \((n - 2) \times 180^\circ\) for each polygon.
1.
Quadrilateral (\( n = 4 \)):
\[
\text{Sum} = (4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]
2.
Pentagon (\( n = 5 \)):
\[
\text{Sum} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
3.
Hexagon (\( n = 6 \)):
\[
\text{Sum} = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ
\]
4.
Heptagon (\( n = 7 \)):
\[
\text{Sum} = (7 - 2) \times 180^\circ = 5 \times 180^\circ = 900^\circ
\]
5.
Octagon (\( n = 8 \)):
\[
\text{Sum} = (8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ
\]
6.
Nonagon (\( n = 9 \)):
\[
\text{Sum} = (9 - 2) \times 180^\circ = 7 \times 180^\circ = 1260^\circ
\]
7.
Decagon (\( n = 10 \)):
\[
\text{Sum} = (10 - 2) \times 180^\circ = 8 \times 180^\circ = 1440^\circ
\]
---
Step 4: Write the Final Answers
The sums of the interior angles for each polygon are:
1. Quadrilateral: \( 360^\circ \)
2. Pentagon: \( 540^\circ \)
3. Hexagon: \( 720^\circ \)
4. Heptagon: \( 900^\circ \)
5. Octagon: \( 1080^\circ \)
6. Nonagon: \( 1260^\circ \)
7. Decagon: \( 1440^\circ \)
---
Final Answer
\[
\boxed{360^\circ, 540^\circ, 720^\circ, 900^\circ, 1080^\circ, 1260^\circ, 1440^\circ}
\]
Parent Tip: Review the logic above to help your child master the concept of interior angles of a polygon worksheet.
