Construct a heptagon and show its exterior angles. Write true or false for the following statements regarding exterior angles: a) For a triangle, there are 6 exterior angles. b) For a convex polygon,

Here are the solutions to the exercises:

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1. Construct a heptagon and show its exterior angles. A heptagon is a polygon with 7 sides. To construct a regular heptagon:

   • Draw a circle.
   • Divide the circle into 7 equal sectors. The angle for each sector at the center would be $\frac{360^\circ}{7} \approx 51.43^\circ$.
   • Connect the points where the sector lines intersect the circle to form the 7 sides of the heptagon.
   • To show the exterior angles, extend each side of the heptagon outwards. The angle formed between the extended side and the adjacent side of the polygon is an exterior angle. There will be 7 exterior angles, one at each vertex.

2. Write true or false for each statement.

   • a) For a triangle, there are 6 exterior angles. A triangle has 3 vertices. At each vertex, there is one distinct exterior angle (formed by extending one side). While you can draw two lines extending from a vertex, the two exterior angles formed are vertically opposite and thus equal. Therefore, a triangle has 3 exterior angles. Answer: False

   • b) For convex polygon, each exterior angle is less than $180^\circ$. In a convex polygon, all interior angles are less than $180^\circ$. Since an interior angle and its corresponding exterior angle sum to $180^\circ$, if the interior angle is less than $180^\circ$, the exterior angle must also be less than $180^\circ$ (and greater than $0^\circ$). Answer: True

   • c) Each exterior angle of a rectangle is $90^\circ$. A rectangle has four interior angles, each measuring $90^\circ$. The exterior angle is found by subtracting the interior angle from $180^\circ$.

     $$180^\circ - 90^\circ = 90^\circ$$

     Answer: True

   • d) If the measure of each interior angle of a regular polygon is $x^\circ$, then the measure of each exterior angle of the given regular polygon is $(360 - x)^\circ$. The sum of an interior angle and its corresponding exterior angle is always $180^\circ$. If the interior angle is $x^\circ$, then the exterior angle is $180^\circ - x^\circ$. Answer: False

Sum of the exterior angles of a polygon

• What can you say about the sum of exterior angles of a polygon? The sum of the exterior angles of any convex polygon is always $\boxed{\text{360}^\circ}$.

• Does the sum depend on the number of sides of the polygon? No, the sum of the exterior angles of a convex polygon does not depend on the number of sides. It is always $360^\circ$, whether it's a triangle (3 sides), a quadrilateral (4 sides), or a heptagon (7 sides).