Question 1168879
Let's break down this geometry problem step by step.

**Understanding the Problem**

* **Regular n-gon:** This means a polygon with 'n' sides, where all sides are equal in length, and all interior angles are equal.
* **A1A2A3...An:** This notation refers to the vertices (corners) of the n-gon, labeled in order.
* **(a) Prove A1A2A3A4 is a trapezoid:** This asks you to show that the quadrilateral formed by the first four vertices of the n-gon has at least one pair of parallel sides.
* **(b) Prove angle A2A1A4=360°/n:** This asks you to find the measure of a specific angle within the n-gon.

**Visualizing the Problem**

It's helpful to draw a diagram. Let's start with a regular hexagon (n=6) to get an idea.

```
      A6
     /  \
    /    \
   A1-----A5
  / \    /
 /   \  /
A2-----A4
 \    /
  \  /
   A3
```

**Part (a): Proving A1A2A3A4 is a Trapezoid**

1.  **Properties of Regular Polygons:**
    * All sides are equal.
    * All interior angles are equal.
    * The measure of each interior angle is (n-2) * 180° / n.

2.  **Angle Relationships:**
    * In a regular n-gon, the arcs between consecutive vertices are equal.
    * Angles subtended by equal arcs are equal.

3.  **Trapezoid Definition:**
    * A trapezoid is a quadrilateral with at least one pair of parallel sides.

4.  **Proof:**
    * Consider the arcs A1A2, A2A3, A3A4, etc. Because we have a regular n-gon, all of these arcs are equal.
    * Angle A2A1An is equal to angle A4A3An-2. This is because they are subtended by the same number of arcs.
    * Because of the equal arcs, and the equal interior angles, the line A1A2, and the line A3A4 are symmetrical in relation to the center of the n-gon.
    * Therefore, the line A1A2 and the line A3A4 are parallel.
    * Since A1A2A3A4 has one pair of parallel sides (A1A2 and A3A4), it is a trapezoid.

**Part (b): Proving Angle A2A1A4=360°/n**

1.  **Central Angles:**
    * The central angle subtended by each side of a regular n-gon is 360° / n.

2.  **Inscribed Angles:**
    * An inscribed angle is an angle formed by two chords in a circle (or in this case, a regular n-gon).
    * The measure of an inscribed angle is half the measure of the intercepted arc (or the central angle that subtends the same arc).

3.  **Proof:**
    * The arc A2A4 subtends a central angle of 2 * (360° / n) because it spans two sides of the n-gon.
    * Angle A2A1A4 is an inscribed angle that intercepts the arc A2A4.
    * Therefore, angle A2A1A4 = (1/2) * [2 * (360° / n)] = 360° / n.

**Key Points**

* Understanding the properties of regular polygons is crucial.
* Visualizing the problem with a diagram helps.
* Remember the relationships between central angles and inscribed angles.

I hope this helps!