Let's go through the questions step by step.

24. What is the measure of each interior angle of a regular decagon?

For a regular polygon, the formula for the measure of each interior angle is:

$\text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n}$

For a decagon (n = 10):

$\text{Interior Angle} = \frac{(10-2) \times 180^\circ}{10} = \frac{8 \times 180^\circ}{10} = 144^\circ$

Thus, the correct answer is c. 144°.

25. Which is NOT a possible angle measure of a regular polygon?

Again, using the formula for the interior angle of a regular polygon, we check which of the given options cannot be the angle for a polygon:

We know the interior angle is:

$\text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n}$

Trying some values:

• $108^\circ$ corresponds to an n = 5 (Pentagon).
• $128.6^\circ$ corresponds to n = 7.
• $135^\circ$ corresponds to n = 8.

But $125^\circ$ does not correspond to a whole number value for n, so b. 125° is not possible.

26. The sum of the measures of the interior angles of a polygon is 1620°. How many sides does the polygon have?

The sum of the interior angles of a polygon is given by:

$\text{Sum of Interior Angles} = (n-2) \times 180^\circ$

Setting this equal to 1620°:

$(n-2) \times 180 = 1620$

Solving for $n$:

$n-2 = \frac{1620}{180} = 9 \implies n = 11$

Thus, the polygon has 11 sides (c. 11).

27. Calculate the measure of $\angle X$ in this isosceles trapezoid.

In an isosceles trapezoid, the two base angles are equal, and the sum of the adjacent angles is 180°. Given one angle as 62°, the adjacent angle $X$ is:

$\angle X = 180^\circ - 62^\circ = 118^\circ$

Thus, the correct answer is d. 118°.

28. Calculate the measure of $\angle X$ in the pentagon below.

The sum of the angles in a pentagon is:

$\text{Sum of Angles} = (5-2) \times 180^\circ = 540^\circ$

Summing the given angles: $110^\circ + 82^\circ + 105^\circ + 153^\circ = 450^\circ$. Therefore:

$\angle X = 540^\circ - 450^\circ = 90^\circ$

None of the choices match, suggesting either a mistake in the question or the figure, but based on calculation, the correct answer should be 90°.

Would you like any details for these explanations? Here are five related questions:

1. How do you derive the formula for the interior angles of any polygon?
2. What is the sum of the interior angles of a hexagon?
3. How do you find the exterior angle of a regular polygon?
4. What is the interior angle of a regular 15-sided polygon?
5. If a polygon has an interior angle sum of 1440°, how many sides does it have?

Tip: For regular polygons, the measure of each exterior angle is easy to calculate: just divide 360° by the number of sides!