Question

1. the table below shows triangles formed when all the diagonals are drawn from one vertex of regular polygons. which of the following statements is true based on the table? table showing number of sides: 3, 4, 5, 6 and number of triangles: 1, 2, 3, 4 a. all the triangles formed in each regular polygon are congruent. b. all the triangles formed in each regular polygon are isosceles. c. the number of triangles formed in any regular polygon is 2 less than the number of sides in the polygon. d. the number of triangles formed in any regular polygon is half the number of sides in the polygon. 6. identify the converse of the statement if cats fly, then ducks bark. a. if cats dont fly, then ducks dont bark. b. if cats bark, then ducks fly. c. if ducks dont bark, then cats dont fly. d. if ducks bark, then cats fly. 7. table showing equation 3(x - 5)=30 with reasons: 3x - 15 = 30 (1. blank), 3x = 45 (2. addition property of equality), x = 15 (3. division property of equality) a. substitution b. multiplication property of equality c. distributive property d. combine like terms

Explanation:

5.

Step1: Recall polygon - triangle relationship

The number of non - overlapping triangles formed by drawing diagonals from one vertex of a polygon with \(n\) sides is \(n - 2\). For a triangle (\(n=3\)), \(3- 2=1\); for a quadrilateral (\(n = 4\)), \(4 - 2=2\); for a pentagon (\(n=5\)), \(5 - 2 = 3\); for a hexagon (\(n = 6\)), \(6-2=4\).

The converse of a conditional statement "If \(p\), then \(q\)" is "If \(q\), then \(p\)". The given statement is "If cats fly, then dogs bark". Here \(p\) is "cats fly" and \(q\) is "dogs bark". The converse is "If dogs bark, then cats fly".

Step1: Analyze the step from \(3(x - 5)=30\) to \(3x-15 = 30\)

The distributive property \(a(b - c)=ab - ac\) is used. Here \(a = 3\), \(b=x\) and \(c = 5\), so \(3(x - 5)=3x-3\times5=3x - 15\).

Answer:

C. The number of triangles formed in any regular polygon is 2 less than the number of sides in the polygon.

6.