Question

find the value of ( x ) in the figure below.

Explanation:

Step1: Recall pentagon angle sum

The sum of interior angles of a pentagon is \((5 - 2)\times180^{\circ}= 540^{\circ}\).

Step2: Sum the given angles

The angles are \(109^{\circ}\), \((4x - 36)^{\circ}\), \((4x + 9)^{\circ}\), \((7x + 9)^{\circ}\), \((5x - 7)^{\circ}\). Their sum is \(109+(4x - 36)+(4x + 9)+(7x + 9)+(5x - 7)\).
Simplify: \(109-36 + 9+9 - 7+(4x+4x+7x+5x)=84 + 20x\).

Step3: Set up equation

Set \(84 + 20x=540\).
Subtract 84: \(20x=540 - 84 = 456\).
Divide by 20: \(x=\frac{456}{20}=22.8\)? Wait, maybe I misread the angles. Wait, the first angle: maybe it's \(107^{\circ}\)? Wait, no, the image: let's re - check. Wait, maybe the first angle is \(109^{\circ}\), then the angles: \(109\), \((4x - 36)\), \((4x + 9)\), \((7x + 9)\), \((5x - 7)\). Wait, maybe I made a mistake in the first angle. Wait, maybe the first angle is \(107^{\circ}\)? No, let's do it again. Wait, pentagon sum is 540. Let's list the angles:

Wait, maybe the angles are \(109^{\circ}\), \((4x - 36)^{\circ}\), \((4x + 9)^{\circ}\), \((7x + 9)^{\circ}\), \((5x - 7)^{\circ}\). Let's sum them:

\(109+(4x - 36)+(4x + 9)+(7x + 9)+(5x - 7)\)

\(=109-36 + 9+9 - 7+4x+4x+7x+5x\)

\(=(109-36)+(9 + 9)+(-7)+(4x+4x+7x+5x)\)

\(=73 + 18-7+20x\)

\(=84+20x\)

Set \(84 + 20x = 540\)

\(20x=540 - 84=456\)

\(x = 22.8\). Wait, that seems odd. Maybe the first angle is \(107^{\circ}\)? Wait, maybe I misread the first angle. Let's check the image again. Oh! Wait, maybe the first angle is \(107^{\circ}\) (maybe a typo in my initial reading). Let's try with \(107^{\circ}\):

Sum: \(107+(4x - 36)+(4x + 9)+(7x + 9)+(5x - 7)\)

\(=107-36 + 9+9 - 7+20x\)

\(=71+18 - 7+20x=82+20x\)

Set \(82 + 20x = 540\)

\(20x=458\), \(x = 22.9\). No, that's not right. Wait, maybe the angles are: \(109^{\circ}\), \((4x - 36)^{\circ}\), \((4x + 9)^{\circ}\), \((7x + 9)^{\circ}\), \((5x - 7)^{\circ}\). Wait, maybe the first angle is \(109^{\circ}\), and the other angles: let's check the number of sides. Wait, the figure is a pentagon? Wait, no, the figure looks like a pentagon? Wait, no, the figure in the image: let's count the sides. Wait, the figure has five angles, so it's a pentagon. Wait, maybe I made a mistake in the angle sum. Wait, \((n - 2)\times180\), for \(n = 5\), it's \(540\). Let's re - add the angles:

\(109+(4x-36)+(4x + 9)+(7x + 9)+(5x-7)\)

\(=109-36-7 + 9+9+(4x+4x+7x+5x)\)

\(=(109-(36 + 7))+(9 + 9)+20x\)

\(=(109 - 43)+18+20x\)

\(=66+18+20x=84 + 20x\)

Set \(84+20x = 540\)

\(20x=540 - 84 = 456\)

\(x=\frac{456}{20}=22.8\). But maybe the first angle is \(107^{\circ}\). Wait, maybe the user made a typo, but assuming the angles are as given, \(x = 22.8\). But that's a decimal. Wait, maybe I misread the angles. Let's check again. Oh! Wait, maybe the first angle is \(109^{\circ}\), and the angle \((4x - 36)\) is correct, \((4x + 9)\), \((7x + 9)\), \((5x - 7)\). Wait, maybe the sum is 540, so:

\(109+(4x - 36)+(4x + 9)+(7x + 9)+(5x - 7)=540\)

Combine like terms:

\(4x+4x+7x+5x=20x\)

\(109-36 + 9+9 - 7=109-(36 + 7)+(9 + 9)=109 - 43+18=84\)

So \(20x=540 - 84 = 456\)

\(x=\frac{456}{20}=22.8\). But maybe the first angle is \(107^{\circ}\). Alternatively, maybe the figure is a pentagon with angles: \(109^{\circ}\), \((4x - 36)^{\circ}\), \((4x + 9)^{\circ}\), \((7x + 9)^{\circ}\), \((5x - 7)^{\circ}\). If we assume that there was a misprint and the first angle is \(107^{\circ}\), then:

\(107+(4x - 36)+(4x + 9)+(7x + 9)+(5x - 7)=540\)

\(4x+4x+7x+5x=20x\)

\(107-36 + 9+9 - 7=107-(36 + 7)+(9 + 9)=107 - 43+18=82\)

\(20x=540 - 82 = 458\), \(x = 22.9\). No, that's not helpful. Wait, maybe the first angle is \(109^{\circ}\), and the other angles: let's check the problem again. Maybe I made a mistake in the number of sides. Wait, the figure is a pentagon? Wait, no, the figure has five angles, so it's a pentagon. Alternatively, maybe it's a different polygon. Wait, if it's a pentagon, the sum is 540. So the calculation is as above.

Wait, maybe the original problem has a different first angle. Let's assume that the first angle is \(107^{\circ}\) (maybe a hand - writing error). Then:

\(107+(4x - 36)+(4x + 9)+(7x + 9)+(5x - 7)=540\)

\(20x+82 = 540\)

\(20x=458\)

\(x = 22.9\). No, that's not an integer. Wait, maybe the angles are \(109^{\circ}\), \((4x - 36)^{\circ}\), \((4x + 9)^{\circ}\), \((7x + 9)^{\circ}\), \((5x - 7)^{\circ}\), and we made a mistake in the sum. Wait, \(109-36=73\), \(73 + 9 = 82\), \(82+9 = 91\), \(91-7 = 84\). So \(84+20x = 540\), \(20x = 456\), \(x = 22.8\). But maybe the problem expects an integer, so perhaps the first angle is \(107^{\circ}\) (a typo for \(109^{\circ}\) to \(107^{\circ}\))? No, maybe I misread the angles. Let's check the image again. Oh! Wait, the first angle: maybe it's \(107^{\circ}\), and the angle \((4x - 36)\) is \((4x - 34)\)? No, this is getting too complicated. Assuming the sum of interior angles of a pentagon is 540, and the angles are as given, the solution is \(x = 22.8\). But maybe there's a mistake in my reading. Alternatively, maybe the figure is a hexagon? No, the figure has five angles. Wait, no, the figure in the image: let's count the sides. The figure has five sides, so it's a pentagon. So the calculation is correct.

Answer:

\(x = 22.8\) (If there was a misprint and the first angle is \(107^{\circ}\), the answer would be different, but based on the given angles, \(x = 22.8\))