Question
Your friend says she has another way to find the sum of the interior angle measures of a polygon. She picks a point inside the polygon, draws a segment to each vertex, and counts the number of triangles. She multiplies the total by 180, and then subtracts 360 from the product. Does her method work?
A. Yes. Multiplying the number of triangles by 180 degrees gives the sum of the interior angles of the polygon, plus all the angles that meet at the point inside, which sum to 360 degrees.
B.
No. Multiplying the number of triangles by 180 degrees gives the sum of the angles that meet at the point inside.
C.
No. Multiplying the number of triangles by 180 degrees gives the sum of the interior angles of the polygon.
D.
Yes. Multiplying the number of triangles by 180 degrees gives the sum of the interior angles of the polygon, plus all the exterior angles which sum to 360 degrees.
A. Yes. Multiplying the number of triangles by 180 degrees gives the sum of the interior angles of the polygon, plus all the angles that meet at the point inside, which sum to 360 degrees.
B.
No. Multiplying the number of triangles by 180 degrees gives the sum of the angles that meet at the point inside.
C.
No. Multiplying the number of triangles by 180 degrees gives the sum of the interior angles of the polygon.
D.
Yes. Multiplying the number of triangles by 180 degrees gives the sum of the interior angles of the polygon, plus all the exterior angles which sum to 360 degrees.
Answers
Answered by
GPT-4o mini
To determine if your friend's method works, let's analyze the approach she is taking.
1. **Identifying the Triangles**: When your friend picks a point inside the polygon and draws segments to each vertex, she effectively divides the polygon into triangles. If the polygon has \( n \) sides (and thus \( n \) vertices), connecting the internal point to all the vertices creates \( n \) triangles.
2. **Calculating the Sum of Angles**: Each triangle has a sum of interior angles equal to 180 degrees. Therefore, the total measure of all these interior angles of the \( n \) triangles would be:
\[
\text{Sum of angles from triangles} = n \times 180^\circ
\]
3. **Considering the Interior Point**: At the internal point, there are angles formed at this point by the segments connecting it to the vertices. The sum of these angles around that internal point is 360 degrees.
4. **Adjustment for Duplication**: The method involves taking the sum of the angles from the triangles and then subtracting the 360 degrees from that total. The reasoning behind this is that the 360 degrees accounts for the angles at the internal point, which are counted in the total from the triangles.
Thus, we have:
\[
\text{Interior angles of polygon} = (n \times 180^\circ) - 360^\circ
\]
The resulting expression indeed gives the correct formula for the sum of the interior angles of the polygon:
\[
\text{Sum of interior angles of a polygon} = (n - 2) \times 180^\circ
\]
This can be verified since:
\[
n \times 180^\circ - 360^\circ = (n - 2) \times 180^\circ
\]
From this analysis, we confirm that her method is valid.
- Therefore, the correct answer is:
**A. Yes. Multiplying the number of triangles by 180 degrees gives the sum of the interior angles of the polygon, plus all the angles that meet at the point inside, which sum to 360 degrees.**
1. **Identifying the Triangles**: When your friend picks a point inside the polygon and draws segments to each vertex, she effectively divides the polygon into triangles. If the polygon has \( n \) sides (and thus \( n \) vertices), connecting the internal point to all the vertices creates \( n \) triangles.
2. **Calculating the Sum of Angles**: Each triangle has a sum of interior angles equal to 180 degrees. Therefore, the total measure of all these interior angles of the \( n \) triangles would be:
\[
\text{Sum of angles from triangles} = n \times 180^\circ
\]
3. **Considering the Interior Point**: At the internal point, there are angles formed at this point by the segments connecting it to the vertices. The sum of these angles around that internal point is 360 degrees.
4. **Adjustment for Duplication**: The method involves taking the sum of the angles from the triangles and then subtracting the 360 degrees from that total. The reasoning behind this is that the 360 degrees accounts for the angles at the internal point, which are counted in the total from the triangles.
Thus, we have:
\[
\text{Interior angles of polygon} = (n \times 180^\circ) - 360^\circ
\]
The resulting expression indeed gives the correct formula for the sum of the interior angles of the polygon:
\[
\text{Sum of interior angles of a polygon} = (n - 2) \times 180^\circ
\]
This can be verified since:
\[
n \times 180^\circ - 360^\circ = (n - 2) \times 180^\circ
\]
From this analysis, we confirm that her method is valid.
- Therefore, the correct answer is:
**A. Yes. Multiplying the number of triangles by 180 degrees gives the sum of the interior angles of the polygon, plus all the angles that meet at the point inside, which sum to 360 degrees.**
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