Let's go through each question on the worksheet in detail and solve them step-by-step:

1. True or False Questions

• A. An Equilateral triangle could also be a Scalene triangle.
  Answer: False. An equilateral triangle has all sides equal, while a scalene triangle has all sides unequal, so they cannot be the same.

• B. An Isosceles triangle could also be a Scalene triangle.
  Answer: False. An isosceles triangle has at least two equal sides, while a scalene triangle has all sides unequal, so they cannot be the same.

• C. An Isosceles triangle could also be a Right triangle.
  Answer: True. An isosceles triangle can have a right angle if the other two angles are each 45 degrees.

• D. An Obtuse triangle could also be a Right triangle.
  Answer: False. An obtuse triangle has one angle greater than 90 degrees, while a right triangle has exactly one 90-degree angle. They cannot be the same.

• E. A polygon must be either concave or convex.
  Answer: True. Every polygon is either concave (having at least one interior angle greater than 180°) or convex (all interior angles less than 180°).

2. Triangle Side Lengths: 12, 26, and X

• To find the possible values of $X$, use the triangle inequality theorem:

  • $X + 12 > 26$
  • $X + 26 > 12$
  • $12 + 26 > X$

  Simplifying each:

  • $X > 14$
  • $X > -14$ (always true, so it doesn't affect the range)
  • $X < 38$

  Possible values for $X$: $14 < X < 38$.

3. Draw Each Shape

• For this question, you would draw:
  • An Isosceles Trapezoid with two parallel sides and two equal-length non-parallel sides.
  • A Rhombus with all sides equal and opposite angles equal.
  • A Square with all sides equal and all angles 90 degrees.

Ensure to mark congruent angles and sides, and label parallel sides as requested.

4. Regular Hexagon with Side Length 24 cm

• To find the radius of a regular hexagon (distance from the center to a vertex), note that in a regular hexagon, the radius is equal to the side length.
  • Radius = 24 cm.

5. Formula for the Sum of Interior Angles of a Polygon

• The formula to find the sum of the interior angles of a polygon with $n$ sides is: $\text{Sum of interior angles} = (n - 2) \times 180^\circ$

• For a 36-sided polygon: $\text{Sum} = (36 - 2) \times 180^\circ = 34 \times 180^\circ = 6120^\circ$

6. Formula for One Interior Angle of a Regular Polygon

• The formula to find one interior angle of a regular polygon with $n$ sides is: $\text{Interior angle} = \frac{(n - 2) \times 180^\circ}{n}$

• For a 24-sided polygon: $\text{Interior angle} = \frac{(24 - 2) \times 180^\circ}{24} = \frac{22 \times 180^\circ}{24} = \frac{3960^\circ}{24} = 165^\circ$

These are the solutions and explanations for each part. Let me know if you'd like more details on any specific question, or if you have any further questions!

Here are some related questions to deepen your understanding:

1. Why can't an equilateral triangle be a scalene triangle?
2. How would the triangle inequality change if we had different side lengths for the triangle?
3. What are the defining properties of a rhombus versus a square?
4. How can you prove that the radius of a regular hexagon is equal to its side length?
5. How would you calculate the exterior angle of a regular 24-sided polygon?

Tip: Remember that the sum of the exterior angles of any polygon is always 360°, regardless of the number of sides.