This geometry maze challenges students to calculate interior and exterior angles of various polygons to find the correct path from start to finish.

To solve this maze, we need to find the value of $x$ (or the specific angle requested) in each polygon. Then, we follow the path where the answer matches the number on the connecting line. We start at the "START!" box and end at the "FINISH!" box.

Here are the key rules we will use:
1. Sum of Interior Angles: For a polygon with $n$ sides, the sum is $(n - 2) \times 180^\circ$.
2. Sum of Exterior Angles: For any convex polygon, the sum of exterior angles is always $360^\circ$.
3. Linear Pair: An interior angle and its adjacent exterior angle add up to $180^\circ$.

Let's go step-by-step through the correct path.

Step 1: The START Box

Shape: Pentagon (5 sides).
Given Interior Angles: $99^\circ$, $94^\circ$, $71^\circ$, $123^\circ$, and $5x^\circ$.
Goal: Find $x$.

1. Calculate the sum of interior angles for a pentagon ($n=5$):
$$ (5 - 2) \times 180 = 3 \times 180 = 540^\circ $$
2. Set up the equation:
$$ 99 + 94 + 71 + 123 + 5x = 540 $$
3. Add the known numbers:
$$ 387 + 5x = 540 $$
4. Subtract 387 from 540:
$$ 5x = 153 $$
5. Divide by 5:
$$ x = 30.6 $$

Wait, looking at the paths out of the START box, the numbers are 20, 17, and 48. My calculation gave 30.6. Let me re-read the angles in the image carefully.
The angles are: $99^\circ$, $94^\circ$, $71^\circ$, $123^\circ$, and $5x^\circ$.
Sum: $99+94+71+123 = 387$.
$540 - 387 = 153$.
$153 / 5 = 30.6$.

Let me look at the other paths. Maybe I misread a number or there is a different starting logic. Let's look at the box to the right of START.
Box Top-Middle: Hexagon (6 sides).
Exterior angles given: $50^\circ$, $48^\circ$, $59^\circ$, $x^\circ$, $x^\circ$, $50^\circ$.
Sum of exterior angles = $360^\circ$.
$$ 50 + 48 + 59 + x + x + 50 = 360 $$
$$ 207 + 2x = 360 $$
$$ 2x = 153 $$
$$ x = 76.5 $$
The path from START to this box is labeled 20. This doesn't seem to match $x$.

Let's look at the box below START.
Box Bottom-Left (connected via 17): Heptagon? No, let's count sides. 1,2,3,4,5,6,7,8. It's an octagon (8 sides).
Angles: $x^\circ, 149^\circ, 2x^\circ, 152^\circ, 183^\circ$ (wait, interior angle > 180? No, that's likely $113^\circ$ or similar. Let's look closer. It looks like $113^\circ$), $123^\circ, 145^\circ, 139^\circ$.
This seems complicated. Let's re-evaluate the START box calculation. Is it possible one of the angles is different?
Ah, looking at the first box again. The angle labeled $123^\circ$ might be something else? Or maybe $71^\circ$?
Let's try working backward from the FINISH box or checking simple integer answers.

Let's look at the box connected to START by the path labeled 17.
Box: Octagon (8 sides).
Sum of interior angles = $(8-2) \times 180 = 6 \times 180 = 1080^\circ$.
Angles: $x, 149, 2x, 152, 113, 123, 145, 139$. (Assuming the blurry one is 113).
Sum of knowns: $149+152+113+123+145+139 = 821$.
Equation: $3x + 821 = 1080 \Rightarrow 3x = 259 \Rightarrow x = 86.33$. Not 17.

Let's look at the box connected to START by the path labeled 48.
Box: Quadrilateral (4 sides).
Angles: $79^\circ, 87^\circ, 105^\circ, x^\circ$.
Sum = $360^\circ$.
$$ 79 + 87 + 105 + x = 360 $$
$$ 271 + x = 360 $$
$$ x = 89 $$
Path label is 48. Answer is 89. No match.

Let's re-examine the START box calculation.
Maybe the angle is not $5x$? It looks like $5x^\circ$.
Maybe the sum is different?
$99+94+71+123 = 387$.
$540-387=153$.
$153/5 = 30.6$.

Is there a path labeled 30.6? No.
Is there a path labeled 31? No.

Let's look at the Top Right Box (Quadrilateral).
Angles: $2x^\circ, (3x+5)^\circ, 2x^\circ, (3x+5)^\circ$.
Sum = $360^\circ$.
$$ 2x + 3x + 5 + 2x + 3x + 5 = 360 $$
$$ 10x + 10 = 360 $$
$$ 10x = 350 $$
$$ x = 35 $$
The path leading TO this box comes from the Hexagon box (Top Middle) with label 40.
The path leading FROM this box goes to FINISH with label 47.

Let's check the Hexagon (Top Middle) again.
Exterior angles: $50, 48, 59, x, x, 50$.
Sum = 360.
$207 + 2x = 360 \rightarrow 2x = 153 \rightarrow x = 76.5$.
The path from START to this box is 20.
The path from this box to the Top Right Quad is 40.
Does $x=76.5$ relate to 40? No.

Let's look at the Middle Right Box (Pentagon).
Exterior angles: $89^\circ, 3x^\circ, 62^\circ, 47^\circ, 47^\circ$.
Sum = 360.
$$ 89 + 3x + 62 + 47 + 47 = 360 $$
$$ 245 + 3x = 360 $$
$$ 3x = 115 $$
$$ x = 38.33 $$
Path from Top Right Quad to this box is 17.
Path from this box to FINISH is 17.
If the answer for this box is related to the path label... wait. The directions say "Answer each question... work through the maze". Usually, the number on the line IS the answer to the previous box.

Let's test this hypothesis: The number on the line exiting a box is the solution for that box.

Hypothesis Test 1: START Box
We calculated $x = 30.6$.
Paths out are 20, 17, 48. None match.

Let me re-read the START box angles.
$99, 94, 71, 123, 5x$.
Is it possible the $123$ is actually $128$?
$99+94+71+128 = 392$. $540-392=148$. $148/5 = 29.6$. No.
Is it possible the $71$ is $76$?
$99+94+76+123 = 392$. Same.
Is it possible the $94$ is $99$?
$99+99+71+123 = 392$. Same.

Let's look at the Bottom Left Box (Octagon).
Path entering is 17. Path exiting is 15.
If the answer to the previous box was 17, we would take this path.
Which box has an answer of 17?

Let's look at the Middle Left Box (Hexagon with exterior angles).
Angles: $x, 49, 59, 50, 49, 84$.
Sum = 360.
$$ x + 49 + 59 + 50 + 49 + 84 = 360 $$
$$ x + 291 = 360 $$
$$ x = 69 $$
Paths into this box: From START (label 17?? No, line from START goes down to this box? No, START connects to Top-Mid, Mid-Left, and Mid-Middle).
Actually, looking at the lines:
START connects to:
1. Top-Middle Hexagon (Label 20)
2. Middle-Left Hexagon (Label 17) -- Wait, the line labeled 17 connects START to the Middle-Left box? Or does it connect the Middle-Left box to somewhere else?
- The line labeled 17 is between START and the Middle-Left Hexagon.
- The line labeled 20 is between START and the Top-Middle Hexagon.
- The line labeled 48 is between START and the Middle-Middle Quadrilateral.

If the rule is "The label on the path is the answer to the box you just left", then:
- For the path labeled 17 to be correct, the START box must have answer 17.
- For the path labeled 20 to be correct, the START box must have answer 20.
- For the path labeled 48 to be correct, the START box must have answer 48.

Let's re-calculate START to see if I can get 17, 20, or 48.
Sum = 540.
Knowns: $99, 94, 71, 123$. Sum = 387.
Remaining = $153$.
Term is $5x$.
$x = 30.6$.

Is it possible the term is not $5x$? Maybe it's $Sx$? No.
Maybe it's $3x$? $153/3 = 51$.
Maybe it's $9x$? $153/9 = 17$. BINGO!
Let's look really closely at the START box angle. It says $5x^\circ$. But if it were $9x^\circ$, the answer would be 17.
Or, perhaps the sum of the known angles is different.
What if the angle labeled $123$ is actually $128$? Sum=392. Rem=148.
What if the angle labeled $71$ is $76$? Sum=392.
What if the angle labeled $94$ is $84$?
$99+84+71+123 = 377$. $540-377=163$.
What if the angle labeled $99$ is $89$?
$89+94+71+123 = 377$.

Let's look at the shape again. It's a pentagon.
Is it possible one of the angles is an exterior angle?
The arc for $123^\circ$ is inside. The arc for $71^\circ$ is inside. The arc for $94^\circ$ is inside. The arc for $99^\circ$ is inside. The arc for $5x^\circ$ is inside.

Let's try a different interpretation. Maybe the question asks for the measure of the angle $5x$, not $x$?
$5x = 153$. No match.

Let's look at the Middle-Left Hexagon (connected by path 17).
If we assume the path 17 is the correct start, then the answer to the START box MUST be 17.
If $x=17$ for the START box, then $5(17) = 85$.
Sum = $99+94+71+123+85 = 472$. This is not 540.
However, if the unknown angle was just $x$ (not $5x$), and $x=153$, no.

Let's look at the Top-Middle Hexagon (connected by path 20).
If the answer to START was 20...
$5(20) = 100$.
Sum = $99+94+71+123+100 = 487$. Not 540.

Let's look at the Middle-Middle Quadrilateral (connected by path 48).
If the answer to START was 48...
$5(48) = 240$. Way too big.

There is a possibility I am misidentifying the starting connections.
START box has three lines coming out of the bottom/right.
1. Down to Middle-Left Hexagon. Label 17.
2. Right to Top-Middle Hexagon. Label 20.
3. Down-Right to Middle-Middle Quad. Label 48.

Let's calculate the answers for the boxes attached to these paths to see if the labels match the *destination* box?
No, mazes usually work by solving the current box to find the exit.

Let's re-read the Top-Middle Hexagon (Exterior Angles).
Angles: $50, 48, 59, x, x, 50$.
Sum = 360.
$207 + 2x = 360 \Rightarrow 2x = 153 \Rightarrow x = 76.5$.
Exit paths from this box:
- To Top-Right Quad: Label 40.
- To Middle-Right Pentagon: Label 47.
- To Middle-Middle Quad: Label 53.
None of these are 76.5.

Let's re-read the Middle-Left Hexagon (Exterior Angles).
Angles: $x, 49, 59, 50, 49, 84$.
Sum = 360.
$x + 291 = 360 \Rightarrow x = 69$.
Exit paths:
- To Bottom-Left Octagon: Label 15.
- To Middle-Middle Quad: Label 36.
- To Start: Label 17.
None are 69.

Let's re-read the Middle-Middle Quadrilateral (Interior Angles).
Angles: $79, 87, 105, x$.
Sum = 360.
$271 + x = 360 \Rightarrow x = 89$.
Exit paths:
- To Start: 48.
- To Middle-Left Hex: 36.
- To Top-Mid Hex: 53.
- To Bottom-Mid Octagon: 8.
- To Middle-Right Pentagon: 22.
- To Bottom-Right Triangle: 30.
None are 89.

Alternative Theory:
Maybe the variable isn't $x$?
In the START box, the angle is $5x^\circ$.
In the Top-Mid Hex, angles are $x^\circ$.
In the Mid-Left Hex, angle is $x^\circ$.

Let's look at the Bottom-Right Triangle.
Angles: $2x^\circ, 77^\circ$ (interior?), and an exterior angle of $105^\circ$?
The diagram shows a triangle.
Interior angles: $2x$, and another one adjacent to exterior $105$?
Let's assume the angle labeled $105^\circ$ is exterior. Then interior is $180-105=75^\circ$.
The angle labeled $77^\circ$ is interior.
The third angle is $2x$? Or is $2x$ exterior?
The arc for $2x$ is outside. So $2x$ is exterior.
The arc for $77$ is inside.
The arc for the third angle is inside.
Exterior angle sum = 360.
Exterior angles: $2x$, $105$, and the exterior supplement of 77 ($180-77=103$).
$2x + 105 + 103 = 360$.
$2x + 208 = 360$.
$2x = 152$.
$x = 76$.
Path into this box is 30. Path out is 66.

Let's look at the Bottom-Middle Octagon.
Interior angles.
$104, 104, 115, 139, 144, 115, 117, 104$?
Let's sum them.
$104+104+115+139+144+115+117+104 = 942$.
Sum for octagon = 1080.
Missing angle? All 8 are listed.
Wait, one is $x$? No, they all look like numbers.
Ah, the top left one is $104^\circ$. Top mid $104^\circ$. Top right $115^\circ$?
Let's look at the label 88.33 on the path leaving this box.
$1080 - 942 = 138$.
If one of those numbers was $x$, and $x=138$...
Or maybe the average? $1080/8 = 135$.

Let's step back. There is often a simpler pattern in these mazes.
Look at the Finish box.
Paths entering Finish:
- From Top-Right Quad: Label 47.
- From Middle-Right Pentagon: Label 17.
- From Bottom-Right Triangle area? Label 13.5? No, that's near the bottom right corner.

Let's solve the Middle-Right Pentagon (Exterior Angles).
Angles: $89, 3x, 62, 47, 47$.
Sum = 360.
$245 + 3x = 360 \Rightarrow 3x = 115 \Rightarrow x = 38.33$.
This doesn't yield an integer.

Let's solve the Top-Right Quadrilateral.
Angles: $2x, 3x+5, 2x, 3x+5$.
Sum = 360.
$10x + 10 = 360 \Rightarrow 10x = 350 \Rightarrow x = 35$.
Path entering: 40.
Path exiting: 47.

Let's solve the Bottom-Left Octagon.
Angles: $x, 149, 2x, 152, 113, 123, 145, 139$.
Sum = 1080.
$3x + 821 = 1080 \Rightarrow 3x = 259 \Rightarrow x = 86.33$.

Let's solve the Bottom-Middle Octagon.
Angles: $104, 104, 115, 139, 144, 115, 117, 104$?
Wait, look at the bottom middle box again.
One angle is $x^\circ$? No.
Let's look at the path label 88.33.
$88.33 \times ? $
$1080 / 12 = 90$.
$1080 / 8 = 135$.

Is it possible the questions ask for the sum of the angles?
Start Box Sum = 540. Path 48? No.

Let's look at the Middle-Left Hexagon again.
$x = 69$.
Path label 15? No.

Let's try one more specific box: The Triangle at the Bottom Right.
Vertices:
1. Bottom Left: Interior $2x$? No, arrow points out. Exterior $2x$.
2. Bottom Right: Interior $77^\circ$.
3. Top: Exterior $105^\circ$.

Using Exterior Angle Theorem:
Sum of exterior angles = 360.
Ext 1: $2x$.
Ext 2: Supplement of 77 = $103$.
Ext 3: $105$.
$2x + 103 + 105 = 360$.
$2x + 208 = 360$.
$2x = 152$.
$x = 76$.

Now, look at the paths around this triangle.
Path entering from Left (from Bottom-Mid Octagon): 30.
Path entering from Top (from Middle-Right Pentagon): 25? No, that's above it.
Path leaving to Right: 66.

If $x=76$, does 76 match any path? No.

Let's reconsider the START box.
What if the angle is not $5x$ but $Sx$? No.
What if the angle labeled $123$ is actually $128$?
$99+94+71+128 = 392$.
$540-392 = 148$.
$148/5 = 29.6$.

What if the angle labeled $99$ is $90$?
$90+94+71+123 = 378$.
$540-378 = 162$.
$162/5 = 32.4$.

What if the angle labeled $94$ is $90$?
$99+90+71+123 = 383$.
$540-383 = 157$.
$157/5 = 31.4$.

What if the angle labeled $71$ is $70$?
$99+94+70+123 = 386$.
$540-386 = 154$.
$154/5 = 30.8$.

Let's look at the path labels again.
From START: 20, 17, 48.
From Top-Mid Hex: 40, 47, 53.
From Mid-Left Hex: 15, 36, 13.
From Mid-Mid Quad: 8, 22, 30, 53, 36, 48.

Let's solve the Top-Middle Hexagon again.
Exterior angles: $50, 48, 59, x, x, 50$.
Sum = 360.
$207 + 2x = 360 \Rightarrow x = 76.5$.
Is there a path labeled 76.5? No.

Let's solve the Middle-Right Pentagon again.
Exterior: $89, 3x, 62, 47, 47$.
Sum = 360.
$245 + 3x = 360 \Rightarrow x = 38.33$.

Let's solve the Bottom-Right Triangle again.
$x = 76$.

Let's solve the Top-Right Quad again.
$x = 35$.

Let's solve the Mid-Mid Quad again.
$x = 89$.

Let's solve the Mid-Left Hex again.
$x = 69$.

Let's solve the Bottom-Left Octagon again.
$x = 86.33$.

Let's solve the Bottom-Mid Octagon.
Sum = 1080.
Angles: $104, 104, 115, 139, 144, 115, 117, 104$?
Sum = 942.
Difference = 138.
If the last angle is $x$, $x=138$.
If the angle labeled 104 is $x$?

Wait! Look at the Bottom-Middle Octagon again.
One of the angles is labeled $x^\circ$?
Looking at the crop, the top-left angle is $104^\circ$. The top-middle is $104^\circ$. The top-right is $115^\circ$?
Actually, looking at the very bottom row, middle box.
The angles are: $104, 104, 115, 139, 144, 115, 117, 104$?
Wait, the angle at the bottom left is $144$?
Let's check the sum of the visible numbers:
$104+104+115+139+144+115+117+104 = 942$.
$1080 - 942 = 138$.
Is there an $x$ in there?
The angle labeled $115$ (top right) looks a bit like $x$? No.
The angle labeled $104$ (bottom left)?

Let's look at the path label 88.33 coming out of the Bottom-Mid Octagon.
$88.33 \times 12 = 1060$?
$88.33 \times 6 = 530$?
$88.33 \times 3 = 265$?

Actually, $1080 / 12.22$?

Let's try a different approach. Follow the integers.

If we assume the Start Box answer is 17 (matching the path label to the left):
Then we go to the Middle-Left Hexagon.
We found $x=69$ for that hexagon.
Does 69 match any outgoing path?
Paths out: 15, 36, 13. No.

If we assume the Start Box answer is 20 (matching the path label up):
Then we go to the Top-Middle Hexagon.
We found $x=76.5$.
Paths out: 40, 47, 53. No.

If we assume the Start Box answer is 48 (matching the path label right):
Then we go to the Middle-Middle Quadrilateral.
We found $x=89$.
Paths out: 8, 22, 30, 53, 36, 48. No.

Is it possible the variables are defined differently?
In the Top-Right Quad: $x=35$.
Path entering is 40. Path exiting is 47.
In the Mid-Right Pentagon: $x=38.33$.
Path entering is 47. Path exiting is 17.
Wait. The path from Top-Right Quad to Mid-Right Pentagon is labeled 17?
No, looking at the grid:
Top-Right Quad is at (Row 1, Col 4).
Mid-Right Pentagon is at (Row 2, Col 3).
The line between them is labeled 17?
Actually, the line labeled 17 is between Mid-Right Pentagon and FINISH.
The line between Top-Right Quad and Mid-Right Pentagon is labeled 47? No, that's vertical.
Let's trace the lines carefully.

Grid Layout:
R1C1: START
R1C2: Top-Mid Hex
R1C3: Top-Right Quad
R1C4: (Empty/Finish is R2C4)

R2C1: Mid-Left Hex
R2C2: Mid-Mid Quad
R2C3: Mid-Right Pentagon
R2C4: FINISH

R3C1: Bot-Left Oct
R3C2: Bot-Mid Oct
R3C3: Bot-Right Tri
R3C4: Bot-Right Pent? (There is a pentagon in the bottom right corner).

Let's solve the Bottom-Right Pentagon (R3C4).
Interior angles: $(3x+5), (6x+5), (3x+5), (3x+5), (3x+5)$?
Labels:
Top: $(6x+5)^\circ$
Right: $(6x+5)^\circ$
Bottom Right: $(3x+5)^\circ$
Bottom Left: $(3x+5)^\circ$
Left: $(3x+5)^\circ$
Sum = 540.
$(6x+5) + (6x+5) + (3x+5) + (3x+5) + (3x+5) = 540$.
$21x + 25 = 540$.
$21x = 515$.
$x = 24.52$.

Path entering from Left (Bot-Right Tri): 66.
Path entering from Top (Mid-Right Pent): 13.5?
Path entering from Top-Right Quad: 47?

This is getting messy. Let's look for a clear integer solution that matches a path label.

Recalculate Top-Right Quad:
$x = 35$.
Is there a path labeled 35? No.

Recalculate Bot-Right Tri:
$x = 76$.
Is there a path labeled 76? No.

Recalculate Mid-Mid Quad:
$x = 89$.
Is there a path labeled 89? No.

Recalculate Mid-Left Hex:
$x = 69$.
Is there a path labeled 69? No.

Recalculate Top-Mid Hex:
$x = 76.5$.

Recalculate Mid-Right Pent:
$x = 38.33$.

Recalculate Bot-Left Oct:
$x = 86.33$.

Recalculate Start:
$x = 30.6$.

Is it possible the question asks for the ANGLE, not x?
Start: Angle $5x = 153$. Path 15? No.
Top-Mid Hex: Angle $x = 76.5$.
Mid-Left Hex: Angle $x = 69$.
Mid-Mid Quad: Angle $x = 89$.
Top-Right Quad: Angle $2x = 70$? Or $3x+5 = 110$?
Bot-Right Tri: Angle $2x = 152$?

Let's look at the Finish box inputs.
1. From Top-Right Quad: Label 47.
2. From Mid-Right Pentagon: Label 17.
3. From Bot-Right Pentagon: Label 13.5.

If the path label is the answer to the PREVIOUS box:
- The Top-Right Quad must have answer 47.
- The Mid-Right Pentagon must have answer 17.
- The Bot-Right Pentagon must have answer 13.5.

Let's check if Mid-Right Pentagon yields 17.
We calculated $x = 38.33$.
What if the question asks for something else?
Sum of interior angles = 540.
What if $x$ is not the variable to solve for?
The angles are $89, 3x, 62, 47, 47$.
If the answer is 17, maybe $3x = 51 \Rightarrow x=17$?
If $x=17$, Sum Ext = $89 + 51 + 62 + 47 + 47 = 296 \neq 360$.

What if the Top-Right Quad yields 47?
We calculated $x=35$.
Angles: $70, 110, 70, 110$.
If the answer is 47... no obvious connection.

Correct Logic for These Mazes:
Usually, you solve for $x$, and the value of $x$ IS the number on the path.
Since none of my $x$ values matched the path labels, I must have misread a number in the START box or the first step.

Let's look at the START box one last time.
$99, 94, 71, 123, 5x$.
Sum = 540.
$387 + 5x = 540$.
$5x = 153$.
$x = 30.6$.

Is it possible the angle is $3x$?
$3x = 153 \Rightarrow x = 51$.
Is it possible the angle is $9x$?
$9x = 153 \Rightarrow x = 17$.

If $x=17$, then the path labeled 17 is the correct one.
This leads to the Middle-Left Hexagon.
Let's assume the path label 17 is correct.
Now we are in the Middle-Left Hexagon.
Exterior angles: $x, 49, 59, 50, 49, 84$.
Sum = 360.
$x + 291 = 360 \Rightarrow x = 69$.
Does 69 match any outgoing path?
Paths: 15, 36, 13.
No.

However, look at the path labeled 13 going to the Bottom-Left Octagon.
And 15 going to the Bottom-Left Octagon?
No, 15 goes from Mid-Left Hex to Bot-Left Oct.
13 goes from Mid-Left Hex to Bot-Mid Oct? No, 13 is between Mid-Left Hex and Bot-Mid Oct?
Actually, the line labeled 13 connects Mid-Left Hex and Bot-Mid Oct.
The line labeled 15 connects Mid-Left Hex and Bot-Left Oct.
The line labeled 36 connects Mid-Left Hex and Mid-Mid Quad.

If the answer to Mid-Left Hex is 69, and no path is 69, this path is dead.

Let's try the path 20 from START.
Leads to Top-Mid Hex.
$x = 76.5$.
Paths: 40, 47, 53.
Dead end.

Let's try the path 48 from START.
Leads to Mid-Mid Quad.
$x = 89$.
Paths: 8, 22, 30, 53, 36, 48.
Dead end.

Conclusion: There is likely a typo in my reading of the START box.
If the angle labeled $123^\circ$ is actually $128^\circ$:
Sum = 392. Rem = 148. $148/5 = 29.6$.
If the angle labeled $71^\circ$ is $76^\circ$:
Sum = 392. Rem = 148.
If the angle labeled $94^\circ$ is $99^\circ$:
Sum = 397. Rem = 143.
If the angle labeled $99^\circ$ is $104^\circ$:
Sum = 392.

What if the polygon is a Hexagon?
No, 5 sides.

What if the sum is not 540?
$(5-2)*180 = 540$. Correct.

I will provide the solution based on the most likely intended path, which often involves integer answers. The only integer answer derived from a standard setup was $x=17$ if the coefficient was 9, or $x=35$ in the Top-Right Quad.

Given the constraints and typical maze designs, the path is likely:
Start -> (Answer 17?) -> ... -> Finish.

Final Answer:
The correct path through the maze is determined by solving for $x$ in each polygon. Based on the calculations, the value of $x$ in the starting pentagon is approximately 30.6, which does not match the exit labels (17, 20, 48) exactly, suggesting a potential ambiguity in the image's numbers (e.g., if the term was $9x$, $x$ would be 17). However, following the standard logic for such problems:

1. Start: Solve for $x$. If we assume the path labeled 17 is correct, we move to the next box.
2. Finish: The final box is reached by solving the preceding polygons.

*(Note: Due to the discrepancy in the starting calculation vs. the provided path labels, the exact numerical path cannot be definitively traced without clearer image data, but the method is to solve for $x$ using the sum of angles formulas.)*

Final Answer:
The problem requires solving for $x$ in each polygon to navigate the maze.
1. Start (Pentagon): Sum of interior angles = $540^\circ$. $99+94+71+123+5x=540 \rightarrow 5x=153 \rightarrow x=30.6$. (No matching path label).
2. Top-Right (Quadrilateral): Sum = $360^\circ$. $2x+(3x+5)+2x+(3x+5)=360 \rightarrow 10x=350 \rightarrow x=35$.
3. Middle-Right (Pentagon): Sum of exterior angles = $360^\circ$. $89+3x+62+47+47=360 \rightarrow 3x=115 \rightarrow x \approx 38.3$.
4. Finish: The maze ends at the "FINISH!" box.

Based on typical integer solutions in such worksheets, if the starting angle was $9x$ instead of $5x$, $x$ would be 17, leading to the left path. Without that correction, the values do not align perfectly with the path labels.