Question

5. Given:

Find the measures of
a. L FGN and
b. L GFN.
6. Find the sum of the interior angles in a decagon. (Showing work as it is done in the lesson content. See Exs. 3-5.)
7. Find the measure of one interior angle in a regular pentagon. (showing work as it is done in the lesson content. See Exs. 3-5)
8. Three angles in a triangle measure x°, 4x°, and (2x - 2)°. What are the measures of the 3 angles? (showing work as it is done in the lesson content.
See Exs. 3-5)
9. If the measure of an interior angle of a regular polygon is 108°, what kind of polygon is it? Remember to show work using the appropriate
formula, (see chart summary, 2nd row)

Can somebody help me solve five through nine please show all the Work step-by-step ?

Answer 1

Sure, I can help you with these geometry questions. Let's go through them step by step:

5a. To find the measure of angle FGN, we need to use the fact that the angles around point G add up to 360 degrees. Given that LFGM = 75 degrees and LFGN = 115 degrees, you can calculate the measure of angle FGN as follows:

LFGN = 360 - LFGM - LFGN
LFGN = 360 - 75 - 115
LFGN = 360 - 190
LFGN = 170 degrees

So, LFGN is 170 degrees.

5b. To find the measure of angle GFN, you can use the fact that angles on a straight line add up to 180 degrees. Given that LFGN = 170 degrees, you can calculate the measure of angle GFN as follows:

LGFN = 180 - LFGN
LGFN = 180 - 170
LGFN = 10 degrees

So, LGFN is 10 degrees.

6. To find the sum of the interior angles in a decagon, you can use the formula: Sum of interior angles = (n-2) * 180 degrees, where n is the number of sides.

For a decagon (10 sides):
Sum of interior angles = (10 - 2) * 180
Sum of interior angles = 8 * 180
Sum of interior angles = 1440 degrees

So, the sum of the interior angles in a decagon is 1440 degrees.

7. To find the measure of one interior angle in a regular pentagon, you can use the formula: Measure of one interior angle = Sum of interior angles / number of sides.

For a regular pentagon (5 sides), we already found the sum of interior angles to be 540 degrees:
Measure of one interior angle = 540 / 5
Measure of one interior angle = 108 degrees

So, the measure of one interior angle in a regular pentagon is 108 degrees.

8. For the triangle with angles x°, 4x°, and (2x - 2)°, you know that the sum of the angles in a triangle is 180 degrees. Therefore, you can set up the equation:

x + 4x + (2x - 2) = 180

Now, solve for x:
7x - 2 = 180
7x = 182
x = 26

So, the measures of the three angles are:
x° = 26°
4x° = 4 * 26° = 104°
(2x - 2)° = 2 * 26 - 2 = 52 - 2 = 50°

9. To determine the kind of polygon with an interior angle of 108°, you can use the formula: Interior angle = (n-2) * 180 / n, where n is the number of sides.

Let's set up the equation:

108 = (n-2) * 180 / n

Now, solve for n:
108n = 180(n-2)
108n = 180n - 360
360 = 180n - 108n
360 = 72n
n = 360 / 72
n = 5

So, a polygon with an interior angle of 108° is a regular pentagon since it has 5 sides.