Find the missing angle in each irregular polygon on this math worksheet.
Worksheet titled "Angles in Irregular Polygon Worksheet" with eight numbered irregular polygons, each showing some angles and one missing angle labeled as x°, asking to find the missing angle.
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Step-by-step solution for: Angles in Polygons Worksheets - Math Monks
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Show Answer Key & Explanations
Step-by-step solution for: Angles in Polygons Worksheets - Math Monks
To find the missing angle in each irregular polygon, we use a simple rule:
The sum of all interior angles in any polygon depends on how many sides it has.
Formula: Sum = (n – 2) × 180°, where n is the number of sides.
We’ll go problem by problem.
---
Problem 1:
This shape has 5 sides → pentagon.
Sum of angles = (5–2) × 180 = 3 × 180 = 540°
Given angles: 90°, 90°, 90°, 154°, and x°
Add known angles: 90 + 90 + 90 + 154 = 424°
So, x = 540 – 424 = 116°
✔ Check: 424 + 116 = 540 → Correct.
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Problem 2:
This shape has 7 sides → heptagon.
Sum = (7–2) × 180 = 5 × 180 = 900°
Given angles: 120°, 118°, 124°, 140°, 148°, 113°, and x°
Add them:
120 + 118 = 238
238 + 124 = 362
362 + 140 = 502
502 + 148 = 650
650 + 113 = 763
x = 900 – 763 = 137°
✔ Check: 763 + 137 = 900 → Correct.
---
Problem 3:
This shape has 7 sides again → heptagon → sum = 900°
Given angles: 102°, 147°, 70°, 128°, 138°, x°, and one more? Wait — let’s count vertices.
Looking at the diagram: labels are 102°, 147°, 70°, 128°, 138°, x°, and that’s 6 angles? But it should be 7 for a heptagon.
Wait — actually, looking again: the shape has 7 corners labeled with angles: 102, 147, 70, 128, 138, x, and... wait, maybe I miscounted.
Actually, counting the labeled angles in Problem 3: there are 7 angles shown: 102°, 147°, 70°, 128°, 138°, x°, and... hold on — the image shows 7 distinct angle marks. Let me list them as given:
From top clockwise: 147°, 70°, 128°, 138°, x°, 102°, and then back to start — that’s 7 angles. So yes, 7 sides.
Sum = 900°
Add knowns: 147 + 70 = 217
217 + 128 = 345
345 + 138 = 483
483 + 102 = 585
So far: 585 + x = 900 → x = 900 – 585 = 315? That can’t be right — too big!
Wait — mistake! I missed one angle? Or did I add wrong?
Let me re-add carefully:
Angles listed: 102°, 147°, 70°, 128°, 138°, x°, and... is there a seventh? Looking at the original image description — actually, in Problem 3, the angles shown are: 102°, 147°, 70°, 128°, 138°, and x° — that’s only 6 angles. But a hexagon has 6 sides.
Ah! I think I made a mistake earlier. Let me recount based on standard interpretation.
In Problem 3: if you look at the shape, it's a hexagon (6 sides). Because typically these worksheets have consistent patterns.
Hexagon: n=6 → sum = (6–2)×180 = 4×180 = 720°
Now, given angles: 102°, 147°, 70°, 128°, 138°, and x° → that’s 6 angles. Perfect.
Add knowns:
102 + 147 = 249
249 + 70 = 319
319 + 128 = 447
447 + 138 = 585
x = 720 – 585 = 135°
✔ Check: 585 + 135 = 720 → Correct.
I initially thought it was 7-sided, but it’s 6-sided. Good catch.
---
Problem 4:
This is a tricky one — it looks like a pentagon with a reflex angle (greater than 180°). The angle marked 282° is inside the shape — so it’s part of the interior angles.
Shape has 5 sides → pentagon → sum = 540°
Given angles: 90°, 90°, 38°, 282°, and x°
Wait — 282° is already over 180°, which is fine for an irregular polygon.
Add knowns: 90 + 90 = 180
180 + 38 = 218
218 + 282 = 500
x = 540 – 500 = 40°
✔ Check: 500 + 40 = 540 → Correct.
Note: Even though 282° is large, it’s still just one interior angle.
---
Problem 5:
This is a hexagon (6 sides) → sum = 720°
Given angles: 160°, 124°, 120°, x°, and two right angles (90° each) — because of the square corner symbols.
So angles: 160, 124, 120, 90, 90, x
Add knowns:
160 + 124 = 284
284 + 120 = 404
404 + 90 = 494
494 + 90 = 584
x = 720 – 584 = 136°
✔ Check: 584 + 136 = 720 → Correct.
---
Problem 6:
This is a pentagon (5 sides) → sum = 540°
Given angles: 140°, x°, 125°, 210°, 32°
Note: 210° is a reflex angle — allowed.
Add knowns:
140 + 125 = 265
265 + 210 = 475
475 + 32 = 507
x = 540 – 507 = 33°
✔ Check: 507 + 33 = 540 → Correct.
---
Problem 7:
Quadrilateral (4 sides) → sum = (4–2)×180 = 360°
Given angles: 125°, 70°, 50°, x°
Add knowns: 125 + 70 = 195
195 + 50 = 245
x = 360 – 245 = 115°
✔ Check: 245 + 115 = 360 → Correct.
---
Problem 8:
Also a quadrilateral → sum = 360°
Given angles: 110°, 115°, 75°, x°
Add knowns: 110 + 115 = 225
225 + 75 = 300
x = 360 – 300 = 60°
✔ Check: 300 + 60 = 360 → Correct.
---
Final Answers:
1. 116°
2. 137°
3. 135°
4. 40°
5. 136°
6. 33°
7. 115°
8. 60°
Final Answer:
116, 137, 135, 40, 136, 33, 115, 60
The sum of all interior angles in any polygon depends on how many sides it has.
Formula: Sum = (n – 2) × 180°, where n is the number of sides.
We’ll go problem by problem.
---
Problem 1:
This shape has 5 sides → pentagon.
Sum of angles = (5–2) × 180 = 3 × 180 = 540°
Given angles: 90°, 90°, 90°, 154°, and x°
Add known angles: 90 + 90 + 90 + 154 = 424°
So, x = 540 – 424 = 116°
✔ Check: 424 + 116 = 540 → Correct.
---
Problem 2:
This shape has 7 sides → heptagon.
Sum = (7–2) × 180 = 5 × 180 = 900°
Given angles: 120°, 118°, 124°, 140°, 148°, 113°, and x°
Add them:
120 + 118 = 238
238 + 124 = 362
362 + 140 = 502
502 + 148 = 650
650 + 113 = 763
x = 900 – 763 = 137°
✔ Check: 763 + 137 = 900 → Correct.
---
Problem 3:
This shape has 7 sides again → heptagon → sum = 900°
Given angles: 102°, 147°, 70°, 128°, 138°, x°, and one more? Wait — let’s count vertices.
Looking at the diagram: labels are 102°, 147°, 70°, 128°, 138°, x°, and that’s 6 angles? But it should be 7 for a heptagon.
Wait — actually, looking again: the shape has 7 corners labeled with angles: 102, 147, 70, 128, 138, x, and... wait, maybe I miscounted.
Actually, counting the labeled angles in Problem 3: there are 7 angles shown: 102°, 147°, 70°, 128°, 138°, x°, and... hold on — the image shows 7 distinct angle marks. Let me list them as given:
From top clockwise: 147°, 70°, 128°, 138°, x°, 102°, and then back to start — that’s 7 angles. So yes, 7 sides.
Sum = 900°
Add knowns: 147 + 70 = 217
217 + 128 = 345
345 + 138 = 483
483 + 102 = 585
So far: 585 + x = 900 → x = 900 – 585 = 315? That can’t be right — too big!
Wait — mistake! I missed one angle? Or did I add wrong?
Let me re-add carefully:
Angles listed: 102°, 147°, 70°, 128°, 138°, x°, and... is there a seventh? Looking at the original image description — actually, in Problem 3, the angles shown are: 102°, 147°, 70°, 128°, 138°, and x° — that’s only 6 angles. But a hexagon has 6 sides.
Ah! I think I made a mistake earlier. Let me recount based on standard interpretation.
In Problem 3: if you look at the shape, it's a hexagon (6 sides). Because typically these worksheets have consistent patterns.
Hexagon: n=6 → sum = (6–2)×180 = 4×180 = 720°
Now, given angles: 102°, 147°, 70°, 128°, 138°, and x° → that’s 6 angles. Perfect.
Add knowns:
102 + 147 = 249
249 + 70 = 319
319 + 128 = 447
447 + 138 = 585
x = 720 – 585 = 135°
✔ Check: 585 + 135 = 720 → Correct.
I initially thought it was 7-sided, but it’s 6-sided. Good catch.
---
Problem 4:
This is a tricky one — it looks like a pentagon with a reflex angle (greater than 180°). The angle marked 282° is inside the shape — so it’s part of the interior angles.
Shape has 5 sides → pentagon → sum = 540°
Given angles: 90°, 90°, 38°, 282°, and x°
Wait — 282° is already over 180°, which is fine for an irregular polygon.
Add knowns: 90 + 90 = 180
180 + 38 = 218
218 + 282 = 500
x = 540 – 500 = 40°
✔ Check: 500 + 40 = 540 → Correct.
Note: Even though 282° is large, it’s still just one interior angle.
---
Problem 5:
This is a hexagon (6 sides) → sum = 720°
Given angles: 160°, 124°, 120°, x°, and two right angles (90° each) — because of the square corner symbols.
So angles: 160, 124, 120, 90, 90, x
Add knowns:
160 + 124 = 284
284 + 120 = 404
404 + 90 = 494
494 + 90 = 584
x = 720 – 584 = 136°
✔ Check: 584 + 136 = 720 → Correct.
---
Problem 6:
This is a pentagon (5 sides) → sum = 540°
Given angles: 140°, x°, 125°, 210°, 32°
Note: 210° is a reflex angle — allowed.
Add knowns:
140 + 125 = 265
265 + 210 = 475
475 + 32 = 507
x = 540 – 507 = 33°
✔ Check: 507 + 33 = 540 → Correct.
---
Problem 7:
Quadrilateral (4 sides) → sum = (4–2)×180 = 360°
Given angles: 125°, 70°, 50°, x°
Add knowns: 125 + 70 = 195
195 + 50 = 245
x = 360 – 245 = 115°
✔ Check: 245 + 115 = 360 → Correct.
---
Problem 8:
Also a quadrilateral → sum = 360°
Given angles: 110°, 115°, 75°, x°
Add knowns: 110 + 115 = 225
225 + 75 = 300
x = 360 – 300 = 60°
✔ Check: 300 + 60 = 360 → Correct.
---
Final Answers:
1. 116°
2. 137°
3. 135°
4. 40°
5. 136°
6. 33°
7. 115°
8. 60°
Final Answer:
116, 137, 135, 40, 136, 33, 115, 60
Parent Tip: Review the logic above to help your child master the concept of geometry polygons worksheet.
