To find the measure of the indicated angles, we use the formula for the sum of interior angles of a polygon:
Sum of interior angles = (n – 2) × 180°,
where
n is the number of sides.
Then, if all angles are equal (regular polygon), divide by n to get each angle.
If not regular, add known angles and subtract from total to find missing ones.
Let’s go one by one.
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Problem 1: Pentagon with angles labeled x, x, x, x, 140°
- Number of sides: 5 → Sum = (5–2)×180 = 3×180 =
540°
- Four angles are x, one is 140° → 4x + 140 = 540
- 4x = 540 – 140 = 400
- x = 400 ÷ 4 =
100°
✔ Each x = 100°
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Problem 2: Hexagon with angles labeled x, x, x, x, x, 120°
- Sides: 6 → Sum = (6–2)×180 = 4×180 =
720°
- Five angles are x, one is 120° → 5x + 120 = 720
- 5x = 720 – 120 = 600
- x = 600 ÷ 5 =
120°
✔ Each x = 120°
Wait — that would mean all angles are 120°, which makes sense for a regular hexagon. But here one is given as 120°, and others are x — so yes, x=120°.
But let me double-check: If it's regular, all angles are 120°, so this fits.
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Problem 3: Pentagon with angles labeled x, x, x, 100°, 110°
- Sides: 5 → Sum = 540°
- Three x’s, plus 100° and 110° → 3x + 100 + 110 = 540
- 3x + 210 = 540
- 3x = 330
- x = 110°
✔ Each x = 110°
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Problem 4: Hexagon with angles labeled x, x, x, x, 90°, 130°
- Sides: 6 → Sum = 720°
- Four x’s, plus 90° and 130° → 4x + 90 + 130 = 720
- 4x + 220 = 720
- 4x = 500
- x = 125°
✔ Each x = 125°
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Problem 5: Pentagon with angles labeled x, x, x, x, 100°
- Sides: 5 → Sum = 540°
- Four x’s, one 100° → 4x + 100 = 540
- 4x = 440
- x = 110°
✔ Each x = 110°
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Problem 6: Hexagon with angles labeled x, x, x, x, x, 100°
- Sides: 6 → Sum = 720°
- Five x’s, one 100° → 5x + 100 = 720
- 5x = 620
- x = 124°
✔ Each x = 124°
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Problem 7: Quadrilateral? Wait — looks like pentagon again? Let me count vertices.
Actually, looking at shape 7: It has 5 sides → pentagon.
Angles: x, x, x, 80°, 100°
- Sum = 540°
- 3x + 80 + 100 = 540
- 3x + 180 = 540
- 3x = 360
- x = 120°
✔ Each x = 120°
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Problem 8: Hexagon with angles labeled x, x, x, x, 110°, 130°
- Sides: 6 → Sum = 720°
- Four x’s, plus 110° and 130° → 4x + 240 = 720
- 4x = 480
- x = 120°
✔ Each x = 120°
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Now, compiling answers:
1. x = 100°
2. x = 120°
3. x = 110°
4. x = 125°
5. x = 110°
6. x = 124°
7. x = 120°
8. x = 120°
Double-checking Problem 6: 5x + 100 = 720 → 5x=620 → x=124 ✔️
Problem 4: 4x + 220 = 720 → 4x=500 → x=125 ✔️
All others check out.
Final Answer:
1. 100°
2. 120°
3. 110°
4. 125°
5. 110°
6. 124°
7. 120°
8. 120°
Parent Tip: Review the logic above to help your child master the concept of sum of interior angles of a polygon worksheet.
