find the indicated angle of each figures with solution

To find the indicated angles for each figure, I'll go through each shape based on the typical properties and formulas related to the angles in polygons. Let's start with each figure:

### 1. **Decagon (10-sided polygon)**
- **Sum of Interior Angles**: 
  \[
  \text{Sum of interior angles} = (n - 2) \times 180^\circ
  \]
  For a decagon (\( n = 10 \)):
  \[
  \text{Sum of interior angles} = (10 - 2) \times 180^\circ = 8 \times 180^\circ = 1440^\circ
  \]
- **Interior Angle of a Regular Decagon**:
  If the decagon is regular (all sides and angles are equal), then:
  \[
  \text{Each interior angle} = \frac{1440^\circ}{10} = 144^\circ
  \]

### 2. **Quadrilateral**
- **Sum of Interior Angles**:
  \[
  \text{Sum of interior angles} = (n - 2) \times 180^\circ
  \]
  For a quadrilateral (\( n = 4 \)):
  \[
  \text{Sum of interior angles} = (4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
  \]
  Depending on the quadrilateral (square, rectangle, etc.), the individual angles can be different.

### 3. **Obtuse Angle Triangle**
- In an obtuse triangle, one of the angles is greater than \( 90^\circ \). The exact angle depends on the specific triangle, but if an angle is marked as obtuse, its value is between \( 90^\circ \) and \( 180^\circ \).

### 4. **Pentagon (5-sided polygon)**
- **Sum of Interior Angles**:
  \[
  \text{Sum of interior angles} = (n - 2) \times 180^\circ
  \]
  For a pentagon (\( n = 5 \)):
  \[
  \text{Sum of interior angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
  \]
- **Interior Angle of a Regular Pentagon**:
  If the pentagon is regular:
  \[
  \text{Each interior angle} = \frac{540^\circ}{5} = 108^\circ
  \]

### 5. **Hexagon (6-sided polygon)**
- **Sum of Interior Angles**:
  \[
  \text{Sum of interior angles} = (n - 2) \times 180^\circ
  \]
  For a hexagon (\( n = 6 \)):
  \[
  \text{Sum of interior angles} = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ
  \]
- **Interior Angle of a Regular Hexagon**:
  If the hexagon is regular:
  \[
  \text{Each interior angle} = \frac{720^\circ}{6} = 120^\circ
  \]

### Summary
1. **Decagon**: Each angle is \(144^\circ\) (if regular).
2. **Quadrilateral**: Sum of angles is \(360^\circ\).
3. **Obtuse Angle Triangle**: One angle is between \(90^\circ\) and \(180^\circ\).
4. **Pentagon**: Each angle is \(108^\circ\) (if regular).
5. **Hexagon**: Each angle is \(120^\circ\) (if regular).

Would you like more details on any of these? Here are some related questions:

1. How do you determine if a polygon is regular?
2. What are the properties of an irregular decagon?
3. How to calculate the exterior angles of a polygon?
4. What are the different types of quadrilaterals and their angle properties?
5. How does the sum of the exterior angles compare across different polygons?

**Tip:** The exterior angle of any regular polygon can be found by dividing 360° by the number of sides.

Learn how to find the indicated angles for various polygons including decagon, quadrilateral, obtuse triangle, pentagon, and hexagon. Detailed solutions and formulas for calculating interior angles of polygons provided. Suitable for Grades 6-8.

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