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 sum of the interior angles of a polygon with k sides can be found using the formula (k-2) * 180 degrees. By substituting the given angles into this formula and solving for k, we can determine the number of sides. In this case, the sum of the given angles is 145 + 157 + (x+15) + (2x-58) + (x+18) + (x-11) + 178 = 7x + 424. Setting this equal to (k-2) * 180 and solving for k, we find that k = 7. Therefore, the value of k is 7.

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
ABCDE is a regular polygon, which means that all of its angles are equal. Since the sum of all the 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 ABCDE is a pentagon, we can substitute n=5 into the formula and solve for each angle. Therefore, each angle in ABCDE measures 108˚.

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
ABCDE is a regular polygon, which means all its angles are equal. In a regular polygon, the sum of all the interior angles is given by the formula (n-2) * 180, where n is the number of sides. Since ABCDE has 5 sides, the sum of its interior angles is (5-2) * 180 = 540˚. Since all angles are equal, each angle measures 540˚/5 = 108˚. Therefore, angle ACD, being one of the angles of the regular polygon, also measures 108˚.

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.