Math worksheet on angles in irregular polygons, featuring problems to calculate missing angles and expressions for sum of interior angles.

Let’s solve each problem step by step.

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Question 1: Write an expression in terms of n, for the sum of the interior angles in any n-sided polygon.

We know that for any polygon with n sides, you can split it into (n - 2) triangles by drawing diagonals from one vertex.

Each triangle has angles that add up to 180°.

So, total sum of interior angles = (n - 2) × 180°

✔ Expression: (n - 2) × 180

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Now let’s fill out the table for questions 2–6.

For each shape:

- Name of Shape → based on number of sides
- Number of sides → count them
- Sum of all angles → use formula: (n - 2) × 180
- Value of missing angle x° → subtract known angles from total sum

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Question 2: Triangle

Shape: Triangle → 3 sides
Sum of angles = (3 - 2) × 180 = 180°

Given angles: 33°, 71°, and x°

So:
x = 180 - 33 - 71 = 76°

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Question 3: Quadrilateral

Shape: Quadrilateral → 4 sides
Sum of angles = (4 - 2) × 180 = 360°

Given angles: 74°, 112°, 96°, and x°

Add known angles: 74 + 112 + 96 = 282°

x = 360 - 282 = 78°

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Question 4: Pentagon

Shape: Pentagon → 5 sides
Sum of angles = (5 - 2) × 180 = 540°

Given angles: 101°, 168°, 126°, 89°, and x°

Add known angles: 101 + 168 = 269; 269 + 126 = 395; 395 + 89 = 484°

x = 540 - 484 = 56°

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Question 5: Hexagon

Shape: Hexagon → 6 sides
Sum of angles = (6 - 2) × 180 = 720°

Given angles: 155°, 163°, 92°, 112°, and a right angle (90°), plus x°

Wait — look at the diagram: there are 6 angles shown? Let’s list them:

From image description (we’re solving logically):

Angles given: 155°, 163°, 92°, 112°, 90° (right angle symbol), and x° → that’s 6 angles → hexagon.

Add known angles:
155 + 163 = 318
318 + 92 = 410
410 + 112 = 522
522 + 90 = 612°

x = 720 - 612 = 108°

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Question 6: Octagon? Wait — let’s count sides/angles carefully.

Looking at the figure: It's a complex irregular polygon. But we can count the number of interior angles marked.

Marked angles:
95°, 209°, 134°, 77°, 84°, 125°, 250°, 87°, 106°, and x° → that’s 10 angles → so it’s a decagon? Wait — no.

Actually, looking again: The shape is drawn as a single closed polygon with indentations (reflex angles). We must count how many vertices (corners) it has.

From the labels:

List of labeled angles:

- 95°
- 209° (reflex)
- 134°
- 77°
- 84°
- 125°
- 250° (reflex)
- 87°
- 106°
- x°

That’s 10 angles → so it’s a 10-gon (decagon)

Sum of interior angles = (10 - 2) × 180 = 8 × 180 = 1440°

Now add all known angles:

Start adding:

95 + 209 = 304
304 + 134 = 438
438 + 77 = 515
515 + 84 = 599
599 + 125 = 724
724 + 250 = 974
974 + 87 = 1061
1061 + 106 = 1167°

Then x = 1440 - 1167 = 273°

Wait — that seems very large. Is that possible?

Yes — because some angles are reflex (greater than 180°), like 209° and 250°. So x could also be reflex.

But let’s double-check addition:

List again:

95
209 → 95+209=304
134 → 304+134=438
77 → 438+77=515
84 → 515+84=599
125 → 599+125=724
250 → 724+250=974
87 → 974+87=1061
106 → 1061+106=1167 ✔

Total sum needed: 1440
x = 1440 - 1167 = 273°

Yes, correct. Even though it’s big, it’s valid for an irregular concave decagon.

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Now compile all answers into the table format requested.

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Final Answer:

1) Expression: (n - 2) × 180

Table:

| # | Name of Shape | Number of sides | Sum of all angles | Value of missing angle x° |
|---|------------------|------------------|--------------------|----------------------------|
| 2 | Triangle | 3 | 180° | 76° |
| 3 | Quadrilateral | 4 | 360° | 78° |
| 4 | Pentagon | 5 | 540° | 56° |
| 5 | Hexagon | 6 | 720° | 108° |
| 6 | Decagon | 10 | 1440° | 273° |

Note: For question 6, even though the shape looks complicated, counting the labeled interior angles gives 10, making it a decagon. Reflex angles (>180°) are allowed in irregular polygons.