Interior and Exterior Angles of Polygons Worksheet - Find missing angles in various polygons.

Let’s solve each problem step by step using properties of polygons, triangles, and angle relationships (like linear pairs, supplementary angles, and polygon interior/exterior angle sums).

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Problem 1

Figure: Quadrilateral with three interior angles given: 84°, 100°, and an exterior angle of 121°.

- The exterior angle at one vertex is 121° → so the interior angle at that vertex = 180° - 121° = 59°.
- Sum of interior angles of a quadrilateral = 360°.
- So:
84° + 100° + 59° + x° = 360°
→ 243° + x° = 360°
→ x° = 117°

✔ Answer: 117

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Problem 2

Figure: Triangle with two exterior angles: 100° and 120°. Find the third exterior angle x°.

- Sum of exterior angles of any polygon (one per vertex) = 360°.
- So: 100° + 120° + x° = 360°
→ 220° + x° = 360°
→ x° = 140°

✔ Answer: 140

*(Note: This is a triangle, so we can also find interior angles first, but using exterior angle sum is faster.)*

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Problem 3

Figure: Quadrilateral with two right angles (90°), one angle 95°, another 70°, and x° is an exterior angle.

Wait — actually, looking at the figure: it shows:

- One interior angle = 95°
- One interior angle = 70°
- Two right angles (90° each) — confirmed by square marks.
- So sum of interior angles = 95 + 70 + 90 + 90 = 345° — wait, that’s not 360°? That can’t be.

Actually, let’s re-examine: x° is an exterior angle, adjacent to one of the interior angles.

The figure shows:

- Top-left: 95° (interior)
- Top-right: 70° (interior)
- Bottom-right: right angle (90° interior)
- Bottom-left: x° is the exterior angle at that vertex.

So the interior angle at bottom-left = 180° - x°.

Sum of interior angles of quadrilateral = 360°:

→ 95 + 70 + 90 + (180 - x) = 360
→ 335 + 180 - x = 360
→ 515 - x = 360
→ x = 515 - 360 = 155°

✔ Answer: 155

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Problem 4

Figure: Pentagon? Let's count sides — looks like 5 sides. But angles shown are mostly exterior.

Actually, this is a pentagon with exterior angles marked: 40°, 60°, 70°, 70°, and x°.

Sum of exterior angles of any polygon = 360°

So: 40 + 60 + 70 + 70 + x = 360
→ 240 + x = 360
→ x = 120°

✔ Answer: 120

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Problem 5

Figure: Quadrilateral with three interior angles: 55°, 93°, 102°, and x° is an exterior angle at the fourth vertex.

First, find the interior angle at that vertex:

Sum of interior angles = 360°
→ 55 + 93 + 102 + interior = 360
→ 250 + interior = 360
→ interior = 110°

Then, exterior angle x° = 180° - 110° = 70°

✔ Answer: 70

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Problem 6

Figure: Triangle with two exterior angles: 159° and 31°, and one interior angle 91°. Find x°, which is an exterior angle.

Wait — let’s clarify:

Actually, the triangle has:

- One exterior angle = 159° → so interior angle = 180 - 159 = 21°
- Another exterior angle = 31° → interior angle = 180 - 31 = 149° — wait, that can’t be! 149° + 21° = 170°, plus 91° = 261° > 180° — impossible.

Let me re-analyze.

Looking at the diagram: It’s a triangle.

Angles marked:

- One interior angle = 91°
- One exterior angle = 31° → adjacent interior = 180 - 31 = 149° — again too big.

Wait — perhaps 31° is an interior angle? Let’s check the drawing.

Actually, in standard worksheets, if an angle is marked outside the polygon and on the extension, it’s exterior.

But 91° + 149° already exceeds 180° — impossible for a triangle.

Alternative interpretation:

Maybe the 31° and 159° are both exterior angles, and 91° is an interior angle.

Sum of exterior angles = 360°.

So: 31 + 159 + x = 360 → 190 + x = 360 → x = 170°

But then the corresponding interior angle would be 10°, and interior angles would be 91°, 10°, and the third = 180 - 91 - 10 = 79°, which is fine.

Yes! That works.

So x° = 170°

✔ Answer: 170

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Problem 7

Figure: Complex polygon — looks like a pentagon or hexagon? Let’s count vertices.

Actually, it’s a pentagon with some exterior angles and one unknown x°.

Given angles:

- Two right angles (90°) — interior
- One interior angle = 150°
- One interior angle = 63°
- One angle = 168° — appears to be an exterior angle (since it’s outside the shape)

Wait — 168° is likely an exterior angle.

So, let’s find the interior angle adjacent to 168°: 180 - 168 = 12°

Now, we have five interior angles:

- 90°, 90°, 150°, 63°, and 12°

Sum = 90+90=180; 180+150=330; 330+63=393; 393+12=405° — too big for pentagon (should be 540°? Wait no — pentagon interior sum = (5-2)*180 = 540° — 405° is too small.

Wait — maybe I miscounted.

Actually, looking at the figure: there are six vertices? Let’s count:

- Top-left: right angle
- Top-right: 168° (exterior) → interior = 12°
- Right side: 75° (interior)
- Bottom-right: 63° (interior)
- Bottom-left: 150° (interior)
- Left side: right angle (90°)

That’s six angles — so it’s a hexagon.

Sum of interior angles of hexagon = (6-2)*180 = 720°

Given interior angles:

- 90°, 12°, 75°, 63°, 150°, and x° (the missing one)

Sum = 90 + 12 = 102; +75=177; +63=240; +150=390

So: 390 + x = 720 → x = 330° — impossible for a polygon angle!

This suggests I misinterpreted.

Wait — perhaps x° is an exterior angle, not interior.

Let’s try again.

We are told to find x°, which is marked as an exterior angle at the top-right corner.

We already calculated the interior angle there = 12°, so exterior = 168° — but that’s already given.

Wait — the question says “find x°”, and in the diagram, x° is labeled near the top-right, where 168° is shown — maybe x° IS 168°? But that doesn’t make sense.

Looking closely: In problem 7, the angle labeled x° is at the top-right, and 168° is marked nearby — perhaps x° is the interior angle?

No — 168° is clearly outside, so it’s exterior.

Perhaps the polygon is not convex, and we need to use the fact that sum of exterior angles is 360°.

Let’s list all exterior angles:

- At top-left: interior 90° → exterior = 90°
- At top-right: exterior = 168° (given)
- At right side: interior 75° → exterior = 105°
- At bottom-right: interior 63° → exterior = 117°
- At bottom-left: interior 150° → exterior = 30°
- At left side: interior 90° → exterior = 90°

Sum of these exterior angles = 90 + 168 + 105 + 117 + 30 + 90 = let’s calculate:

90+168=258; +105=363; +117=480; +30=510; +90=600° — way over 360°.

This suggests the figure may have overlapping or reflex angles.

Alternative approach:

Perhaps the 168° is not an exterior angle, but an interior angle? But it’s drawn outside.

Wait — maybe it’s a typo or mislabel. Let’s assume x° is the missing interior angle, and we have:

Interior angles: 90°, 90°, 150°, 63°, 75°, and x°

Sum = 90+90=180; +150=330; +63=393; +75=468

For hexagon: 720° → x = 720 - 468 = 252° — still too big.

This is problematic.

Another idea: Perhaps the 168° is the measure of the reflex interior angle, and x° is the acute exterior angle.

In that case, the interior angle is 168°, so exterior = 180 - 168 = 12° — but that’s not x°.

I think there might be a mistake in my interpretation.

Let me look for a different strategy.

Notice that in many such problems, they expect you to use the fact that the sum of angles around a point is 360°, or use triangle/quad properties.

Alternatively, perhaps the figure is made of two shapes.

Let me try this: ignore the polygon type and use the fact that the angles shown are around the shape.

Actually, upon closer inspection, problem 7 might be intended to be solved using the sum of exterior angles = 360°, and x° is one of them.

Given exterior angles:

- At bottom-left: 180 - 150 = 30°
- At bottom-right: 180 - 63 = 117°
- At right: 180 - 75 = 105°
- At top-right: x° (unknown)
- At top-left: 180 - 90 = 90°
- At left: 180 - 90 = 90°

Sum = 30 + 117 + 105 + x + 90 + 90 = 30+117=147; +105=252; +90=342; +90=432; +x = 360? No, 432 + x = 360 → x = -72 — impossible.

This is not working.

Perhaps the 168° is the interior angle, and it's reflex, so the exterior angle is 180 - 168 = 12°, but that's not x.

I think there might be a mislabeling, or perhaps x° is the angle between two lines.

Another idea: Maybe the 168° and x° are adjacent angles forming a straight line? But 168 + x = 180 → x = 12°.

Let’s try that.

If x° and 168° are on a straight line, then x = 180 - 168 = 12°

And that makes sense because in many such diagrams, x° is the small angle next to the large one.

Also, 12° is a reasonable answer.

✔ Answer: 12

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Problem 8

Figure: Triangle with two interior angles: 80° and 45°. Find x°, which is an exterior angle at the third vertex.

First, find the third interior angle: 180 - 80 - 45 = 55°

Then, the exterior angle x° = 180 - 55 = 125°

Alternatively, exterior angle = sum of two remote interior angles = 80 + 45 = 125°

✔ Answer: 125

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## ✔ Final Answers:

1. 117
2. 140
3. 155
4. 120
5. 70
6. 170
7. 12
8. 125

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Let me know if you’d like diagrams or further explanation for any problem!