**Question: Suppose a convex polygon has exactly 4 obtuse interior angles. What is the maximum number of the sides of such a polygon?**

**Hints behind the information**

**Convex polygon？**A polygon with all its interior angles less than 180°**exactly 4 obtuse interior angles:**90°<obtuse angle<180° (**H**int: the other interior anlges ≤ 90°)**maximum number of the sides?**What’s the relationship between**sides and interior angles?**

Let’s use regular polygons to explore as all the regular polygons are convex polygons.

**Visual Observation**: **sides and interior angles**

**Observation 1:**

- The
**More**the sides, the**Greater**the interior angles. - The
**Greater**the interior angles, the**More**the sides. - The number of sides is the same as the number interior angles.

**Observation 2:**

- If there are
**4 right angles**, it is a**rectangle**. - This indicates that when there are
**more than 4 sides**, there are**3 right angles at most**.(refer to the graph above, by cutting a right angle, there are 3 right angles left and the polygon turns from a square into a pentagon.)

**Observation 3:**

- Based on “Observation 1” and the graphs above, if there are 3 right angles, the more the sides, the greater the interior angles.

**Conclusion about sides and interior angles:**

So far, we can conclude that to get the **maximum **number of sides, the interior **angles** should be as **great** as possible.

**Combine the conclusion & information：**

**exactly 4 obtuse interior angles**: All the other angles**= 90°**allow for the maximum number of sides.- There are
**3 right angles at most**for a polygon with more than 4 sides.

**4 obtuse angles + 3 right angles = 7 angels –> 7 sides**

**Use formula and inequality **

**The sum of interior angles = (n-2)∙180° **

This sum should be less than the sum under the maximum situation. But what is the maximum situation?

**exactly 4 obtuse interior angles implies:**

- the sum of the 4 obtuse angles < 4∙180° (all angles in a polygon is less than 180°)
- the sum of the other angles ≤ (n-4)∙90° (the other angles are either acute angles or right angles, a maximum situation occurs when all the other angles are 90°. )

**Inequality:** **Sum < the maximum situation**

**(n-2)∙180° < 4∙180°+(n-4)∙90°****n < 8****So, n(max)=7**

This polygon has **7 sides** at most. It can be a **heptagon**.

**Test your evolution**

A convex polygon with 4n+2 sides can be expressed as A_{1}A_{2}A_{3}—A_{(4n+2)}. If each interior angle is a multiple of 30 degrees and**∠𝐴 _{1}=∠𝐴_{2}=∠𝐴_{3}=90°**, find the value of n.

**Can you use the above-mentioned methods to solve this question?**

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