To solve this problem, we need to find the value of [tex]\( x \)[/tex] and subsequently determine the measures of each interior angle of the given pentagon.
### Step-by-Step Solution:
1. Identify the interior angles:
The angles of the pentagon are given as:
[tex]\[
(x-5)^\circ, (x-6)^\circ, (2x-7)^\circ, x^\circ, (2x-2)^\circ
\][/tex]
2. Recall the formula for the sum of interior angles of a pentagon:
The sum of the interior angles of a polygon with [tex]\( n \)[/tex] sides is given by:
[tex]\[
\text{Sum of interior angles} = (n-2) \times 180^\circ
\][/tex]
For a pentagon ([tex]\( n = 5 \)[/tex]):
[tex]\[
\text{Sum of interior angles} = (5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\][/tex]
3. Set up the equation:
The sum of the given angles should equal the sum of the interior angles of the pentagon:
[tex]\[
(x-5) + (x-6) + (2x-7) + x + (2x-2) = 540
\][/tex]
4. Combine like terms:
[tex]\[
x + x + 2x + x + 2x - 5 - 6 - 7 - 2 = 540
\][/tex]
[tex]\[
7x - 20 = 540
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
7x - 20 = 540
\][/tex]
[tex]\[
7x = 560
\][/tex]
[tex]\[
x = 80
\][/tex]
6. Calculate each interior angle:
- First angle: [tex]\( (x-5)^\circ = (80-5)^\circ = 75^\circ \)[/tex]
- Second angle: [tex]\( (x-6)^\circ = (80-6)^\circ = 74^\circ \)[/tex]
- Third angle: [tex]\( (2x-7)^\circ = (2 \times 80 - 7)^\circ = 153^\circ \)[/tex]
- Fourth angle: [tex]\( x^\circ = 80^\circ \)[/tex]
- Fifth angle: [tex]\( (2x-2)^\circ = (2 \times 80 - 2)^\circ = 158^\circ \)[/tex]
### Summary:
- The value of [tex]\( x \)[/tex] is [tex]\( 80 \)[/tex].
- The measures of each interior angle of the pentagon are:
- [tex]\( 75^\circ, 74^\circ, 153^\circ, 80^\circ, 158^\circ \)[/tex]
This completes the solution.
