Question

6.2 assignment
find the value of x.
(there is a geometric figure with angles 90°, 60°, 2x°, x°, and 4x°)
x =

Explanation:

Step1: Sum of angles in a pentagon? No, it's a polygon with angles around? Wait, actually, the figure is a polygon (maybe a pentagon? Wait, no, the angles given: 90°, 60°, 4x°, x° (wait, no, the 2x° and the adjacent angle? Wait, actually, the sum of the interior angles of a pentagon? Wait, no, the figure has angles: 90°, 60°, 4x°, and then the other two angles: 2x° and its adjacent angle which is x°? Wait, no, actually, when dealing with a polygon (maybe a pentagon) or using the fact that the sum of angles around a point? Wait, no, the correct approach is that the sum of the interior angles of a pentagon is (5 - 2)*180 = 540°? Wait, no, let's check the angles: 90°, 60°, 4x°, 2x°, and the angle adjacent to 2x° which is 180° - x° (since they are supplementary). Wait, yes! Because the angle marked 2x° and the angle x° are on a straight line, so they are supplementary. So the angle inside the polygon is 180° - x°. Now, sum of interior angles of a pentagon (5 sides) is (5 - 2)*180 = 540°. So let's list the angles: 90°, 60°, 4x°, 2x°, and (180 - x)°. Wait, no, wait the figure: let's see the angles: 90°, 60°, 4x°, and then the two angles: 2x° and the angle opposite? Wait, maybe I made a mistake. Wait, actually, the correct way is that the sum of the angles in the figure (which is a pentagon? Or a polygon with 5 angles) should be 360°? No, no, (n - 2)*180. Wait, let's count the number of sides. The figure has 5 sides? Wait, the angles given: 90°, 60°, 4x°, 2x°, and the angle adjacent to x° (which is 180 - x°). So sum of angles: 90 + 60 + 4x + 2x + (180 - x) = 540? Wait, no, (5 - 2)*180 = 540. Let's compute: 90 + 60 = 150; 4x + 2x - x = 5x; 180. So total: 150 + 5x + 180 = 540. Then 330 + 5x = 540. 5x = 210? No, that can't be. Wait, maybe it's a quadrilateral? Wait, no, the figure has 5 angles? Wait, maybe I misread. Wait, the angles are: 90°, 60°, 4x°, 2x°, and x°? No, the 2x° and x° are supplementary? Wait, no, the angle marked 2x° and the angle x° are on a straight line, so they are supplementary, so the interior angle is 180 - x. Wait, maybe the figure is a pentagon, but let's check again. Wait, another approach: the sum of the exterior angles of any polygon is 360°, but no, these are interior angles. Wait, maybe the figure is a quadrilateral? Wait, no, the angles: 90°, 60°, 4x°, and the angle adjacent to 2x° (which is 180 - x) and 2x°? Wait, I think I made a mistake. Let's start over.

Wait, the correct formula: for a polygon with n sides, sum of interior angles is (n - 2)*180. Let's count the number of sides. The figure has 5 sides? Let's see: the angles are 90°, 60°, 4x°, 2x°, and the angle opposite to x° (which is 180 - x°). So n = 5. So sum is (5 - 2)*180 = 540. So:

90 + 60 + 4x + 2x + (180 - x) = 540

Simplify:

90 + 60 = 150; 180 = 180; so 150 + 180 = 330; 4x + 2x - x = 5x; so 330 + 5x = 540

Subtract 330: 5x = 210; x = 42? No, that doesn't seem right. Wait, maybe the figure is a quadrilateral (4 sides). Then sum is (4 - 2)*180 = 360. Let's check: angles are 90°, 60°, 4x°, and the angle adjacent to 2x° (180 - x) and 2x°? No, 4 sides: 90, 60, 4x, and (180 - x + 2x)? No, that's not. Wait, maybe the figure is a pentagon but I miscounted. Wait, another way: the angles given are 90°, 60°, 4x°, 2x°, and x°? No, the 2x° and x° are supplementary, so the interior angle is 180 - x. Wait, maybe the problem is using the sum of angles in a polygon (maybe a pentagon) as 360? No, that's for a quadrilateral. Wait, I think I made a mistake. Let's look at the angles again: 90°, 60°, 4x°, 2x°, and the angle adjacent to x° (which is 180 - x°). So sum: 90 + 60 + 4x + 2x + (180 - x) = 90 + 60 + 180 + 4x + 2x - x = 330 + 5x. Now, if it's a pentagon, sum is 540, so 330 + 5x = 540 → 5x = 210 → x = 42. But that seems high. Wait, maybe the figure is a quadrilateral (4 sides), so sum is 360. Then 90 + 60 + 4x + (180 - x) + 2x? No, that's 5 angles. Wait, maybe the figure is a pentagon but the sum is 360? No, that's not. Wait, maybe the problem is about a cyclic polygon? No, the user provided a figure with angles 90°, 60°, 4x°, 2x°, and x° (with 2x° and x° supplementary). Wait, let's try again. The sum of the angles should be 360? No, (5 - 2)*180 = 540. Wait, maybe the figure is a pentagon, so:

90 + 60 + 4x + 2x + (180 - x) = 540

90 + 60 = 150; 180 = 180; 150 + 180 = 330; 4x + 2x - x = 5x; 330 + 5x = 540; 5x = 210; x = 42. But let's check with x = 30. Wait, maybe I messed up the angle. Wait, another approach: the angle marked 2x° and x° are vertical angles? No, they are on a straight line, so supplementary. Wait, maybe the figure is a quadrilateral, so sum is 360. Let's assume it's a quadrilateral: angles are 90°, 60°, 4x°, and (180 - x) + 2x? No, that's 5 angles. Wait, I think the correct way is that the sum of the angles in the figure (which is a pentagon) is 540, so:

90 + 60 + 4x + 2x + (180 - x) = 540

Simplify:

90 + 60 = 150

180 = 180

150 + 180 = 330

4x + 2x - x = 5x

330 + 5x = 540

5x = 540 - 330 = 210

x = 210 / 5 = 42. Wait, but let's check with x = 30. If x = 30, then 4x = 120, 2x = 60, 180 - x = 150. Then sum: 90 + 60 + 120 + 60 + 150 = 480, which is not 540. If x = 42, 4x = 168, 2x = 84, 180 - 42 = 138. Sum: 90 + 60 + 168 + 84 + 138 = 90+60=150; 168+84=252; 138. 150+252=402; 402+138=540. Yes, that works. Wait, but maybe the figure is a different polygon. Wait, maybe the angle adjacent to 2x° is not 180 - x, but x? No, because they are on a straight line, so they must be supplementary. So the correct answer is x = 42? Wait, but let's check again. Wait, maybe I made a mistake in the number of sides. Let's count the sides: the figure has 5 sides, so it's a pentagon, sum of interior angles is 540. So the angles are 90°, 60°, 4x°, 2x°, and (180 - x)°. So adding them up: 90 + 60 + 4x + 2x + 180 - x = 540. Which simplifies to 330 + 5x = 540 → 5x = 210 → x = 42. So that's the solution.

Step2: Solve for x.

From the equation \( 330 + 5x = 540 \), subtract 330 from both sides:
\( 5x = 540 - 330 \)
\( 5x = 210 \)
Divide both sides by 5:
\( x = \frac{210}{5} \)
\( x = 42 \)

Answer:

42