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Let's go over the questions from the image you uploaded. The topic is "Classifying Polygons and Perimeter and Area of a Polygon," and the questions cover various topics related to polygons and their properties. Here's how we can approach them:

### Section I: Find the value of the angles in these polygons
Each polygon has angles marked, and you're asked to find the unknown angle labeled \(a^\circ\). The sum of the interior angles of a polygon is calculated by the formula:

\[
\text{Sum of interior angles} = (n - 2) \times 180^\circ
\]

Where \(n\) is the number of sides of the polygon. After finding the sum of interior angles, subtract the given angles to find the unknown \(a^\circ\).

1. **Figure (a)**: Pentagon (5 sides)
   \[
   \text{Sum of interior angles} = (5 - 2) \times 180^\circ = 540^\circ
   \]
   Sum of known angles: \(110^\circ + 110^\circ + 95^\circ + 85^\circ = 400^\circ\)

   \[
   a^\circ = 540^\circ - 400^\circ = 140^\circ
   \]

2. **Figure (b)**: Hexagon (6 sides)
   \[
   \text{Sum of interior angles} = (6 - 2) \times 180^\circ = 720^\circ
   \]
   Sum of known angles: \(140^\circ + 95^\circ + 125^\circ + 115^\circ = 475^\circ\)

   \[
   a^\circ = 720^\circ - 475^\circ = 245^\circ
   \]

3. **Figure (c)**: Quadrilateral (4 sides)
   \[
   \text{Sum of interior angles} = (4 - 2) \times 180^\circ = 360^\circ
   \]
   Sum of known angles: \(145^\circ + 160^\circ + 30^\circ = 335^\circ\)

   \[
   a^\circ = 360^\circ - 335^\circ = 25^\circ
   \]

4. **Figure (d)**: Quadrilateral (4 sides)
   \[
   \text{Sum of interior angles} = 360^\circ
   \]
   Sum of known angles: \(60^\circ + 40^\circ + 90^\circ = 190^\circ\)

   \[
   a^\circ = 360^\circ - 190^\circ = 170^\circ
   \]

5. **Figure (e)**: Pentagram (5 sides)
   For a star-shaped polygon, we need to sum the external angles, which should equal 360°.

### Section II: Find the number of sides for a regular polygon with a given interior angle
The formula to find the number of sides, \(n\), when given the interior angle of a regular polygon is:

\[
\text{Interior angle} = \frac{(n - 2) \times 180^\circ}{n}
\]

Solving this equation for \(n\), use the given angles from the quiz:

1. For an interior angle of \(162^\circ\):
   \[
   162 = \frac{(n - 2) \times 180}{n}
   \]

### Section III: Naming polygons based on their sides

You need to match the number of sides to the polygon names.

### Section IV: Problem solving involving areas and perimeters of polygons

We can discuss specific questions based on your needs. Would you like to go into more detail for any specific section or question from the quiz? Feel free to ask!

### Expanding on the concepts:
1. How do you calculate the area of a regular hexagon?
2. How do you find the perimeter of a regular polygon given its side length?
3. Can you explain how central angles in regular polygons are related to their number of sides?
4. What is the difference between regular and irregular polygons?
5. How do you calculate the exterior angles of a polygon?

**Tip**: For regular polygons, the sum of the exterior angles is always 360°, regardless of the number of sides.

Plane and Solid Mensuration Quiz No.1 focused on classifying polygons and finding perimeter and area of a polygon. It includes multiple problems requiring calculation of unknown angles in polygons, the number of sides in polygons from given interior angles, and other properties.

Explore a detailed guide on solving problems related to the classification of polygons, calculation of unknown angles, and determining sides based on interior angles. This guide is ideal for high school students looking to understand geometric properties of polygons.

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