Angles Of Polygons

Explanation
A k-sided polygon has k interior angles. In this case, we are given that three of the interior angles measure 145, 157, and 178 degrees. We are also told that the remaining angles measure 150 degrees each. To find the value of k, we need to determine how many angles measure 150 degrees. Since the sum of the interior angles in a polygon is given by (k-2) * 180 degrees, we can set up the equation (k-2) * 180 = 145 + 157 + 178 + (k-3) * 150. Simplifying this equation, we find that k = 13. Therefore, the value of k is 13.

Explanation
The correct answer is 40˚ because in a regular nonagon, the sum of all exterior angles is always 360˚. Since there are 9 exterior angles in a nonagon, each exterior angle would be 360˚ divided by 9, which equals 40˚.

Explanation
A regular hexagon has six sides that are all equal in length and six angles that are all equal in measure. To find the measure of each interior angle, we can use the formula (n-2) * 180, where n is the number of sides of the polygon. In this case, n is 6, so (6-2) * 180 = 4 * 180 = 720. Since all angles are equal, we divide 720 by 6 to find that each interior angle is 120˚.

Explanation
A regular pentagon has 5 equal sides and 5 equal interior angles. To find the measure of each angle, you need to know that the sum of the interior angles of a pentagon is 540 degrees. Divide this sum by the number of angles (5) to find the measure of each angle: 540° / 5 = 108°. Therefore, angle ABC in a regular pentagon measures 108 degrees.

Explanation
The answer is 140˚ because in a regular polygon, all angles are equal. Since the sum of all angles in a polygon is equal to (n-2) * 180˚, where n is the number of sides, we can calculate the measure of each angle by dividing the sum by the number of sides. In this case, if we assume that the polygon has 9 sides, the sum of all angles would be (9-2) * 180˚ = 1260˚. Dividing this sum by 9, we get 140˚ for each angle.

Explanation
A regular hexagon has 6 equal sides and 6 equal interior angles. To find the measure of each interior angle, you can use the formula: (n-2) * 180° / n, where 'n' is the number of sides. For a hexagon, this is (6-2) * 180° / 6 = 120°.
However, the question asks for angle ACD, which is not an interior angle. To find this, we need to draw a diagonal from A to C, creating an isosceles triangle ABC. Each base angle of this triangle (angles BAC and BCA) measures half the difference between the interior angle and 180°: (120° - 180°) / 2 = 30°.
Since the angles in a triangle add up to 180°, angle ACD is 180° - 30° - 30° = 120°. Finally, we divide this by 2 to find the measure of angle ACD: 120° / 2 = 72°.

Explanation
Since the sum of the interior angles of a k-sided polygon is given by the formula (k-2) * 180 degrees, we can set up the equation (k-2) * 180 = 145 + 157 + (x+15) + (2x-58) + (x+18) + (x-11) + x + 178. Simplifying the equation gives us 180k - 360 = 540 + 6x. Rearranging the equation further, we get 6x = 180k - 900. Since both x and k are integers, we can see that the value of x must be a multiple of 6. The only answer choice that satisfies this condition is 106, so it is the correct answer.

Explanation
The sum of all interior angles in an octagon is equal to 1080 degrees. We can set up an equation to find the value of x by adding up all the given angles and setting it equal to 1080. By simplifying the equation, we can solve for x and find that x is equal to 110.

Explanation
The sum of the interior angles of a nonagon is given by the formula (n-2) * 180, where n is the number of sides of the polygon. In this case, n=9, so the sum of the interior angles is (9-2) * 180 = 7 * 180 = 1260˚.

We are given that three of the interior angles measure 158˚, 115˚, and 123˚. To find the value of x, we subtract the sum of these three angles from the total sum of the interior angles: 1260˚ - (158˚ + 115˚ + 123˚) = 864˚.

Since the remaining angles are each equal to (x + 15)˚, we can set up the equation 3(x + 15) = 864˚. Solving for x gives us x = 129.