An illustration shows two triangles A B C. One triangle is labeled figure A, and the other triangle is labeled figure B. For the triangle in figure A, side A C forms the base, and B is the vertex on the top. A closed circle is at the center of the triangle below the vertex B. Three lines are drawn from the circle to the vertices of A, B, and C. For the triangle in figure B, the triangle A B C has a vertical perpendicular line drawn from the middle of side A C to vertex B, dividing it into 2 smaller equal triangles inside the larger triangle. The smaller triangle on the left is labeled A, and the smaller triangle on the right is labeled B.

The description you've provided indicates that both figures A and B represent decompositions of triangle ABC into smaller triangles.

• Figure A shows a triangle with a circle at the centroid, from which lines are drawn to the vertices, suggesting it has been divided into three triangles (one at each vertex).

• Figure B describes a triangle with a vertical perpendicular line from the midpoint of side AC to vertex B, dividing it into two smaller equal triangles.

Since both figures effectively demonstrate a way to decompose triangle ABC into smaller triangles, the correct choice would be:

3 for both figure A and B.

An illustration shows two dodecagons. The dodecagon on the left is labeled figure 1, and the dodecagon on the right is labeled figure 2. Starting at the bottom left vertex of both figures, the vertices are labeled A, B, C, D, E, F, G, H, I, J, K, and L. In both figures, lines connect the vertices F and H, F and I, F and J, F and K, F and L, F and A, F and B, and F and C.

select the correct decomposition of the dodecagon into n - 2 triangles

the correct decomposition is figure __

To decompose a dodecagon (12-sided polygon) into ( n - 2 ) triangles, where ( n ) is the number of sides, we can use the triangulation method.

For a dodecagon, ( n = 12 ), so we need to decompose it into ( 12 - 2 = 10 ) triangles.

From your description, it seems that both figure 1 and figure 2 involve the connections of vertex F to various other vertices, which indicates that they might both be valid approaches to triangulating the dodecagon.

However, without seeing the specific illustrations in figure 1 and figure 2, I can't definitively select between them. Generally, if both figures maintain a proper triangulation by only connecting vertices without crossing lines and result in the correct number of triangles, then either might be correct.

If any of the figures do something like connect vertices in a way that overlaps or creates non-triangular shapes, it needs to be assessed accordingly.

Based on your description, if both figures provide valid connections that yield 10 triangles, then the correct answer would depend on which one you find matches a proper triangulation best.

If you have to select one:

• If both decompose the dodecagon correctly into 10 triangles each without overlap or errors, then you may need additional information or a detailed visual to determine which one to select.
• If only one clearly shows an appropriate triangulation, select that one.

Without further specifics about the figures, you can try to identify if either figure is incorrect in terms of triangulation or configuration. If they both appear to triangulate correctly, choose figure 1 or figure 2 based on which one seems more accurate to your understanding.

An illustration of a pentagon with the vertices labeled upper A, upper B, upper C, upper D, and upper E clockwise from the bottom.

find the sum of the interior angles of the pentagon

the sum of the interior angles of the polygon is __ degrees

To find the sum of the interior angles of a polygon, you can use the formula:

[ \text{Sum of interior angles} = (n - 2) \times 180^\circ ]

where ( n ) is the number of sides in the polygon.

For a pentagon, ( n = 5 ):

[ \text{Sum of interior angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ ]

Therefore, the sum of the interior angles of the pentagon is 540 degrees.