Question

Find the sum of the measures of the angles of a decagon.


A.
2160°


B.
1800°


C.
1440°


D.
900°

Answer 1

Use the Polygon Angle-Sum Theorem to find the measure of an unknown angle in a polygon.

The Polygon Angle-Sum Theorem
The sum of the measures of the angles of an n-gon is (n−2)180.

Giant Panda 1

What is the measure of ∠D in ABCD?


A.
90°


B.
93°


C.
95°


D.
105°

Answer 2

B. 1800°

The sum of the measures of the angles in a decagon is equal to (n-2) * 180, where n is the number of sides in the polygon. In this case, n = 10.

(10-2) * 180 = 8 * 180 = 1440°

Answer 3

To find the measure of angle ∠D in triangle ABCD, we need to first determine how many sides the polygon has. Since this is a quadrilateral (4 sides), we can use the Polygon Angle-Sum Theorem formula:

Sum of Interior Angles = (n-2) * 180, where n is the number of sides in the polygon.

For a quadrilateral (n=4):
Sum of Interior Angles = (4-2) * 180 = 2 * 180 = 360 degrees

Now, we know that the sum of the interior angles of the quadrilateral ABCD is 360 degrees. Given that the other three angles A, B, and C are labeled as 85°, 80°, and 92° respectively, we can find ∠D:

Sum of Angles A, B, C, and D = 360 degrees

85° + 80° + 92° + ∠D = 360 degrees

(85° + 80° + 92°) + ∠D = 360 degrees

257° + ∠D = 360 degrees

∠D = 360 degrees - 257 degrees

∠D = 103 degrees

Therefore, the measure of angle ∠D in quadrilateral ABCD is 103°.

The closest option provided is D, 105°.