2. Numerical completion: \( \begin{array}{l}\text { a) A convex octagon has diagonals. } \\ \text { b) The sum of the angles in a convex octagon is In a regular octagon, each angle measures degrees. } \\ \text { 3. Complete the statement with always, sometimes, or never: } \\ \text { a) A rhombus is the radius of a circle is } 6.4 \mathrm{~cm} \text {, then the diameter is parallelogram. } \\ \text { b) A rectangle is a square. } \\ \text { c) A trapezoid is a parallelogram. } \\ \text { d) A square is a rectangle. }\end{array} \)

Let's break down the problem step by step. ### Part 2: Numerical Completion **a)** A convex octagon has diagonals. To find the number of diagonals in a convex octagon, we can use the formula for the number of diagonals \( D \) in a polygon with \( n \) sides: \[ D = \frac{n(n-3)}{2} \] For an octagon, \( n = 8 \): \[ D = \frac{8(8-3)}{2} = \frac{8 \times 5}{2} = \frac{40}{2} = 20 \] So, a convex octagon has **20 diagonals**. **b)** The sum of the angles in a convex octagon is: The sum of the interior angles \( S \) of a polygon with \( n \) sides is given by the formula: \[ S = (n-2) \times 180 \] For an octagon, \( n = 8 \): \[ S = (8-2) \times 180 = 6 \times 180 = 1080 \text{ degrees} \] In a regular octagon, each angle measures: \[ \text{Each angle} = \frac{S}{n} = \frac{1080}{8} = 135 \text{ degrees} \] ### Part 3: Complete the Statement **a)** A rhombus is a parallelogram. - **Answer:** **Always**. A rhombus is defined as a type of parallelogram where all sides are of equal length. **b)** A rectangle is a square. - **Answer:** **Sometimes**. A rectangle can be a square if all its sides are equal, but not all rectangles are squares. **c)** A trapezoid is a parallelogram. - **Answer:** **Never**. A trapezoid has at least one pair of parallel sides, but it is not necessarily a parallelogram, which requires both pairs of opposite sides to be parallel. **d)** A square is a rectangle. - **Answer:** **Always**. A square is a special type of rectangle where all sides are equal. ### Summary of Answers 1. **a)** 20 diagonals 2. **b)** The sum of the angles is 1080 degrees; each angle in a regular octagon measures 135 degrees. 3. **Complete the statements:** - a) Always - b) Sometimes - c) Never - d) Always