Find the measure of each interior and exterior angle of a regular hexagon and a regular octagon.

Here are the solutions for Exercise 6.8: Exercise 6.8 1. Find the measure of each interior and exterior angle of: i) a regular hexagon A regular hexagon has n=6 sides. Step 1: Calculate the measure of each interior angle. The formula for each interior angle of a regular polygon is ((n-2) × 180^)/(n). Interior angle = ((6-2) × 180^)/(6) = (4 × 180^)/(6) = (720^)/(6) = 120^ Step 2: Calculate the measure of each exterior angle. The formula for each exterior angle of a regular polygon is (360^)/(n). Exterior angle = (360^)/(6) = 60^ Alternatively, the exterior angle is 180^ - interior angle. Exterior angle = 180^ - 120^ = 60^ Answer: Each interior angle is 120^ and each exterior angle is 60^. ii) a regular octagon A regular octagon has n=8 sides. Step 1: Calculate the measure of each interior angle. Interior angle = ((8-2) × 180^)/(8) = (6 × 180^)/(8) = (1080^)/(8) = 135^ Step 2: Calculate the measure of each exterior angle. Exterior angle = (360^)/(8) = 45^ Alternatively, the exterior angle is 180^ - interior angle. Exterior angle = 180^ - 135^ = 45^ Answer: Each interior angle is 135^ and each exterior angle is 45^. 2. What is the sum of each interior and its corresponding exterior angle of the above two regular polygons? The sum of an interior angle and its corresponding exterior angle at any vertex of a polygon is always 180^. For a regular hexagon: 120^ + 60^ = 180^ For a regular octagon: 135^ + 45^ = 180^ Answer: The sum of each interior and its corresponding exterior angle for both polygons is 180^. 3. A Mathematics teacher gave a group work for students to construct a convex regular polygon with one of the interior angle measures 135^. What is the name of this polygon? Step 1: Find the measure of each exterior angle. The sum of an interior angle and its corresponding exterior angle is 180^. Exterior angle = 180^ - Interior angle = 180^ - 135^ = 45^ Step 2: Find the number of sides (n) of the polygon. The formula for each exterior angle of a regular polygon is (360^)/(n). 45^ = (360^)/(n) n = (360^)/(45^) = 8 Step 3: Name the polygon. A polygon with 8 sides is an octagon. Answer: The name of this polygon is a regular octagon. 4. For n-sided regular polygon, the measure of each interior angle is 5 times the measure of each exterior angle. What is the number of sides of this polygon? Let I be the interior angle and E be the exterior angle. Step 1: Set up equations based on the given information. We are given I = 5E. We also know that I + E = 180^. Step 2: Solve for the exterior angle E. Substitute I = 5E into the second equation: 5E + E = 180^ 6E = 180^ E = (180^)/(6) = 30^ Step 3: Find the number of sides (n). The formula for each exterior angle of a regular polygon is E = (360^)/(n). 30^ = (360^)/(n) n = (360^)/(30^) = 12 Answer: The number of sides of this polygon is 12. 5. The diagram shows a regular hexagon and a regular octagon. Calculate the size of the angle marked x. Step 1: Find the interior angle of the regular hexagon. From question 1 i), the interior angle of a regular hexagon is 120^. Step 2: Find the interior angle of the regular octagon. From question 1 ii), the interior angle of a regular octagon is 135^. Step 3: Calculate angle x. The angles around a point sum to 360^. The angle x along with the interior angles of the hexagon and octagon form a complete circle at their common vertex. 120^ + 135^ + x = 360^ 255^ + x = 360^ x = 360^ - 255^ x = 105^ Answer: The size of the angle marked x is 105^. That's 2 down. 3 left today — send the next one.