https://file.aiquickdraw.com/mathbot/file/1726782644445s4gabpsn.jpg

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!

Multiple-choice math problems involving interior angles of polygons, trapezoids, and pentagons, along with solving for unknown angles in given shapes.

Step-by-step solutions to problems involving the measure of interior angles in polygons like decagons, isosceles trapezoids, and pentagons. Learn to apply key geometric formulas and theorems to solve for unknown angles, suitable for students in Grades 7-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation