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.