Question

6.2 assignment
possible points: 0.67
find the value of x.
(there is a diagram with angles: 90°, 60°, 4x°, 2x°, and x°)
x =

Explanation:

Step1: Sum of angles in a polygon (hexagon? No, it's a pentagon? Wait, actually, the figure is a polygon with angles 90°, 60°, 4x°, x° (vertical angle), and 2x°? Wait, no, the sum of exterior angles? Wait, no, the sum of angles around a point? No, the figure is a polygon where the sum of interior angles? Wait, actually, the sum of the angles in a pentagon? Wait, no, let's count the angles. The given angles are 90°, 60°, 4x°, and then we have a linear pair: 2x° and x°? Wait, no, the sum of the angles in a polygon (let's see, the figure has 5 angles? Wait, no, the sum of the interior angles of a pentagon is (5 - 2)*180 = 540? Wait, no, maybe it's a different approach. Wait, actually, the sum of the angles around a point is 360? No, the figure is a polygon with angles 90°, 60°, 4x°, and then two angles: 2x° and its adjacent angle which is x° (vertical angle? Wait, no, the angle adjacent to 2x° is x°, so they are supplementary? Wait, no, 2x° and x°: wait, the angle marked 2x° and the angle marked x° are vertical angles? No, vertical angles are equal, but here 2x° and x°: maybe it's a linear pair? Wait, no, let's re-examine.

Wait, the correct approach: the sum of the interior angles of a pentagon is (5 - 2)*180 = 540°? Wait, no, the figure has angles: 90°, 60°, 4x°, and then two angles: one is 2x°, and the other is x° (since they are vertical angles? No, vertical angles are equal, so that can't be. Wait, maybe the angle adjacent to 2x° is x°, so they are supplementary? Wait, no, 2x + x = 180? No, that would be 3x = 180, x = 60, but that doesn't fit. Wait, maybe the sum of the angles in the figure is 360°? Wait, no, let's list all angles: 90°, 60°, 4x°, 2x°, and x° (since the angle opposite to x° is equal? No, maybe the sum of these angles is 360? Wait, 90 + 60 + 4x + 2x + x = 360? Wait, 90 + 60 is 150, so 150 + 7x = 360? 7x = 210, x = 30? Wait, no, that doesn't seem right. Wait, maybe the sum of the angles is 360°? Wait, let's check again.

Wait, the figure is a polygon with angles: 90°, 60°, 4x°, and then two angles: 2x° and x° (since they are adjacent and form a linear pair? No, 2x and x: if they are adjacent, then 2x + x = 180, so 3x = 180, x = 60, but then 4x = 240, 90 + 60 + 240 + 120 + 60 = 570, which is more than 360. So that's wrong.

Wait, maybe the sum of the exterior angles? No, exterior angles sum to 360. Wait, no, the correct approach: the sum of the interior angles of a quadrilateral is 360°? Wait, the figure: let's count the sides. If it's a quadrilateral, then (4 - 2)180 = 360. So angles: 90°, 60°, 4x°, and then the fourth angle: which is 180 - 2x (since 2x and its adjacent angle are supplementary) and 180 - x? No, this is confusing. Wait, let's look at the diagram again. The diagram has a 90° angle, a 60° angle, a 4x° angle, a 2x° angle, and an x° angle (vertical angle to the angle adjacent to 2x°). Wait, maybe the sum of all these angles is 360°? Wait, 90 + 60 + 4x + 2x + x = 360? Let's calculate: 90 + 60 = 150, 4x + 2x + x = 7x, so 150 + 7x = 360. Then 7x = 210, x = 30. Wait, but let's check: 90 + 60 + 430 + 2*30 + 30 = 90 + 60 + 120 + 60 + 30 = 360. Yes! So that works.

Step1: Sum of angles is 360°

The angles given are 90°, 60°, 4x°, 2x°, and x°. So:
$$90 + 60 + 4x + 2x + x = 360$$

Step2: Simplify the left side

Combine like terms:
$$150 + 7x = 360$$

Step3: Solve for x

Subtract 150 from both sides:
$$7x = 360 - 150$$
$$7x = 210$$

Step4: Divide by 7

$$x = \frac{210}{7}$$
$$x = 30$$

Answer:

30