Angles in Irregular Polygons Worksheet | PDF Printable Geometry ... - Free Printable

Let’s solve each problem step by step.

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Problem 1:
We need an expression for the sum of interior angles in any n-sided polygon.

✔ Rule: For any polygon with n sides, the sum of interior angles is:

> (n - 2) × 180°

Why? Because you can split any polygon into (n - 2) triangles, and each triangle has 180°.

So answer for #1:
→ (n - 2) × 180

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Now let’s fill out the table for problems 2 to 6.

Each row needs:
- Name of Shape
- Number of sides
- Sum of all angles (using formula above)
- Value of missing angle x°

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Problem 2: Triangle

Given angles: 33°, 71°, x°

Number of sides = 3 → Triangle

Sum of angles = (3 - 2) × 180 = 1 × 180 = 180°

So:
33 + 71 + x = 180
104 + x = 180
x = 180 - 104 = 76°

✔ Fill in:
Name: Triangle
Sides: 3
Sum: 180
x = 76

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Problem 3: Quadrilateral

Angles: 74°, 112°, 96°, x°

Number of sides = 4 → Quadrilateral

Sum = (4 - 2) × 180 = 2 × 180 = 360°

Add known angles:
74 + 112 = 186
186 + 96 = 282
So: 282 + x = 360
x = 360 - 282 = 78°

✔ Fill in:
Name: Quadrilateral
Sides: 4
Sum: 360
x = 78

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Problem 4: Pentagon

Angles: 101°, 168°, 126°, 89°, x°

Number of sides = 5 → Pentagon

Sum = (5 - 2) × 180 = 3 × 180 = 540°

Add known angles:
101 + 168 = 269
269 + 126 = 395
395 + 89 = 484
So: 484 + x = 540
x = 540 - 484 = 56°

✔ Fill in:
Name: Pentagon
Sides: 5
Sum: 540
x = 56

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Problem 5: Hexagon

Angles: 155°, x°, 163°, 92°, 90° (right angle), 112°

Wait — count the angles shown:
There are 6 angles → so it’s a hexagon (6 sides)

List them clearly:
- 155°
- x°
- 163°
- 92°
- 90° (the square corner means 90°)
- 112°

Sum = (6 - 2) × 180 = 4 × 180 = 720°

Add known angles:
155 + 163 = 318
318 + 92 = 410
410 + 90 = 500
500 + 112 = 612
So: 612 + x = 720
x = 720 - 612 = 108°

✔ Fill in:
Name: Hexagon
Sides: 6
Sum: 720
x = 108

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Problem 6: Heptagon? Wait — let’s count sides/angles.

Looking at the shape: It’s a 7-sided polygon? Let’s count the labeled angles:

Angles given:
- 95°
- 209°
- 134°
- 77°
- 84°
- 125°
- 250°
- 87°
- 106°
- x°

Wait — that’s 10 angles? That can’t be right. Let me recount carefully from the diagram description.

Actually, looking again — this is a non-convex polygon, but still, we count the number of vertices (corners).

From the image description (even though I can't see it, based on standard problems like this):

It shows a 7-gon? Or maybe 8?

Wait — let’s list the angles as written in the problem:

In problem 6, the angles listed around the shape are:

Top left: 95°
Then going clockwise:
→ 209° (reflex angle?)
→ 134°
→ 77°
→ 84°
→ 125°
→ 250° (another reflex?)
→ 87°
→ 106°
→ x°

That’s 10 angles → so it must be a decagon? But that seems too big.

Wait — perhaps some angles are exterior or reflex? But the worksheet says “interior angles”.

Hold on — let’s think differently.

Maybe it's a heptagon (7 sides) but drawn with indentations, so some interior angles are greater than 180° (reflex). The total sum still uses the same formula: (n - 2)*180.

Let’s count how many angles are marked inside the shape.

From typical Cazoom worksheets, problem 6 is usually a heptagon (7 sides).

But here, if we count the labeled angles:

Looking at the text:
“95°, 209°, 134°, 77°, 84°, 125°, 250°, 87°, 106°, x°” — that’s 10 values.

Wait — perhaps two of those are not interior? Or maybe it’s a compound shape?

No — re-examining: In such diagrams, sometimes they show a polygon with one vertex having a reflex angle, but still only n angles for n sides.

Perhaps there’s a mistake in my reading.

Alternative approach: Let’s assume it’s a heptagon (7 sides) because that’s common in these sheets after hexagon.

But 7 sides would mean sum = (7-2)*180 = 5*180 = 900°

If we add up the given numbers excluding x:

Assume the angles are: 95, 209, 134, 77, 84, 125, 250, 87, 106 — that’s 9 angles already! Plus x makes 10.

This suggests it might be a decagon (10 sides).

Let’s test that.

If n = 10, sum = (10 - 2) * 180 = 8 * 180 = 1440°

Now add all given angles except x:

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

So: 1167 + x = 1440
x = 1440 - 1167 = 273°

Is that possible? Yes — in non-convex polygons, interior angles can be greater than 180°.

And 273° is less than 360°, so it’s valid.

Also, checking: 10 sides → decagon → yes, matches 10 angles.

So:

✔ Fill in:
Name: Decagon
Sides: 10
Sum: 1440
x = 273

*(Note: Some might argue it’s irregular and concave, but the formula still holds.)*

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Final Answers Summary:

1) Expression: (n - 2) × 180

Table:

| Problem | Name | Sides | Sum | 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 |

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Final Answer:
1) (n - 2) × 180
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