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

Let’s solve each problem step by step.

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

Think about it:
- A triangle (3 sides) has angle sum = 180°
- A quadrilateral (4 sides) can be split into 2 triangles → 2 × 180° = 360°
- A pentagon (5 sides) → 3 triangles → 3 × 180° = 540°

So, for an n-sided polygon, you can always split it into (n - 2) triangles.

Therefore, the formula is:

> Sum = (n - 2) × 180°

✔ Final Answer for Q1: (n - 2) × 180

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Question 2: Triangle with angles 33°, 71°, and x°

Triangle → 3 sides → Sum of angles = 180°

Add known angles:
33 + 71 = 104

Subtract from total:
x = 180 - 104 = 76

Name of shape: Triangle
Number of sides: 3
Sum of all angles: 180
Value of missing angle x°: 76

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Question 3: Quadrilateral with angles 74°, 112°, 96°, and x°

Quadrilateral → 4 sides → Sum = (4 - 2) × 180 = 360°

Add known angles:
74 + 112 + 96 = let’s compute:

74 + 112 = 186
186 + 96 = 282

x = 360 - 282 = 78

Name of shape: Quadrilateral
Number of sides: 4
Sum of all angles: 360
Value of missing angle x°: 78

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Question 4: Pentagon with angles 101°, 168°, 126°, 89°, and x°

Pentagon → 5 sides → Sum = (5 - 2) × 180 = 540°

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

x = 540 - 484 = 56

Name of shape: Pentagon
Number of sides: 5
Sum of all angles: 540
Value of missing angle x°: 56

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Question 5: Hexagon with angles 155°, x°, 163°, 92°, and one right angle (90°), and 112°

Wait — let’s count the angles shown:

From diagram:
- 155°
- x°
- 163°
- 92°
- 90° (right angle symbol)
- 112°

That’s 6 angles → hexagon → 6 sides → Sum = (6 - 2) × 180 = 720°

Add known angles:
155 + 163 = 318
318 + 92 = 410
410 + 90 = 500
500 + 112 = 612

x = 720 - 612 = 108

Name of shape: Hexagon
Number of sides: 6
Sum of all angles: 720
Value of missing angle x°: 108

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Question 6: Octagon? Let’s count the sides/angles.

Looking at the figure: It has 8 vertices → octagon → 8 sides → Sum = (8 - 2) × 180 = 1080°

Angles given:
95°, 134°, 77°, 84°, 87°, 106°, 250°, 209°, and x° — wait, that’s 9 angles? That can’t be.

Wait — look again. The shape is drawn as a self-intersecting or complex polygon? But actually, counting the outer corners:

Actually, looking carefully — this is an octagon (8 sides). The angles listed are:

Top left: 95°
Top: 134°
Top right: 77°
Right side: 84°
Bottom right: 87°
Bottom: 106°
Left bottom: x°
And two reflex angles inside: 209° and 250° — but those are NOT interior angles of the polygon! Wait — no, in irregular polygons, especially concave ones, some interior angles can be greater than 180°.

But let’s count how many angles are marked:

Marked angles:
- 95°
- 134°
- 77°
- 84°
- 87°
- 106°
- 209°
- 250°
- x° ← that’s 9 angles? That would mean 9 sides?

Wait — perhaps I miscounted.

Actually, looking at the drawing: it's a star-like shape? No — it’s a simple octagon with two “dents” making two reflex angles.

Count the vertices: starting from top-left going clockwise:

1. 95°
2. 134°
3. 77°
4. 84°
5. 87°
6. 106°
7. x°
8. Then there’s a vertex between 106° and x°? Actually, the 209° and 250° are also interior angles — they’re just reflex angles.

So total angles: 8 angles.

List them:

- 95°
- 134°
- 77°
- 84°
- 87°
- 106°
- 209°
- 250°
- and x° — that’s 9? That doesn’t make sense.

Wait — maybe the 209° and 250° are not separate — let me re-express.

Actually, looking at the image description: the shape has 8 sides. The angles labeled are:

At the points:
- Top: 134°
- Upper right: 77°
- Right: 84°
- Lower right: 87°
- Bottom: 106°
- Lower left: x°
- Left: 95°
- And then two large angles inside the dents: 209° and 250° — but those are part of the same polygon.

Actually, if you trace the perimeter, you’ll find 8 vertices. So 8 interior angles.

The angles given are:

1. 95°
2. 134°
3. 77°
4. 84°
5. 87°
6. 106°
7. 209°
8. 250°
and x° is replacing one of them? No — x° is one of the eight.

Wait — in the diagram, x° is at the lower-left corner, and the other seven are labeled: 95, 134, 77, 84, 87, 106, and then two more: 209 and 250 — that’s nine labels? That must be a mistake.

Actually, upon closer inspection — the 209° and 250° are the reflex angles at the "indentations", and x° is another angle. But total should be 8 angles for octagon.

Perhaps the 209° and 250° are included in the list, and x° is the eighth.

Let me assume the eight angles are:

- 95°
- 134°
- 77°
- 84°
- 87°
- 106°
- 209°
- 250°
— that’s eight, but where is x°? In the diagram, x° is clearly marked at one vertex, so probably one of these is x°? No — the problem says “value of missing angle x°”, so x° is unknown, and others are given.

Looking back at user’s image description: for question 6, angles are: 95°, 134°, 77°, 84°, 87°, 106°, 209°, 250°, and x° — that’s nine values. That suggests 9 sides? But visually it looks like 8.

Wait — perhaps it’s a non-simple polygon, but still, number of sides equals number of angles.

I think there might be a mislabeling, but let’s count the sides in the drawing described:

Starting from top:
- Side 1 to 134° vertex
- Side 2 to 77°
- Side 3 to 84°
- Side 4 to 87°
- Side 5 to 106°
- Side 6 to x°
- Side 7 to 95°
- Side 8 back to start — but then what about 209° and 250°? Those must be at the indentations, meaning they are additional vertices.

Actually, in a concave octagon, you can have reflex angles, but still only 8 vertices.

I suspect that the 209° and 250° are two of the eight angles, and x° is the third unknown? But the table asks for "value of missing angle x°", implying only one missing.

Perhaps the diagram has 8 angles, and seven are given numerically, one is x°.

Let me list the angles as per standard interpretation of such problems:

In many textbooks, for such a shape, the angles are:

- 95°
- 134°
- 77°
- 84°
- 87°
- 106°
- 209°
- 250°
— and x° is not among them? That doesn't make sense.

Wait — looking at the original problem statement: for question 6, the shape has angles labeled: 95°, 134°, 77°, 84°, 87°, 106°, 209°, 250°, and x° — that’s nine labels. This must be a mistake, or perhaps it's a 9-gon?

Count the sides: if you go around the shape, you hit 9 vertices? Let's simulate:

Start at top-left: 95°
Then up to 134°
Then to 77°
Then down to 84°
Then to 87°
Then to 106°
Then to x°
Then to 250° (reflex)
Then to 209° (reflex)
Then back to start — that’s 9 vertices.

Oh! So it’s a nonagon (9 sides)!

Yes, that makes sense. Some irregular polygons can have many sides.

So, 9 sides → Sum = (9 - 2) × 180 = 7 × 180 = 1260°

Now add all given angles except x°:

Given: 95, 134, 77, 84, 87, 106, 209, 250

Add them step by step:

95 + 134 = 229
229 + 77 = 306
306 + 84 = 390
390 + 87 = 477
477 + 106 = 583
583 + 209 = 792
792 + 250 = 1042

Total sum needed: 1260°

So x = 1260 - 1042 = 218

Is that possible? Yes, since it's a concave polygon, angles can be greater than 180°.

So:

Name of shape: Nonagon
Number of sides: 9
Sum of all angles: 1260
Value of missing angle x°: 218

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Now compiling all answers:

Final Answers:

1) Expression: (n - 2) × 180

2) Name: Triangle, Sides: 3, Sum: 180, x = 76

3) Name: Quadrilateral, Sides: 4, Sum: 360, x = 78

4) Name: Pentagon, Sides: 5, Sum: 540, x = 56

5) Name: Hexagon, Sides: 6, Sum: 720, x = 108

6) Name: Nonagon, Sides: 9, Sum: 1260, x = 218

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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) Nonagon, 9, 1260, 218