Interior Angles, Exterior Angles and the Sum | Polygon Worksheets - Free Printable
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Step-by-step solution for: Interior Angles, Exterior Angles and the Sum | Polygon Worksheets
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Show Answer Key & Explanations
Step-by-step solution for: Interior Angles, Exterior Angles and the Sum | Polygon Worksheets
Let’s solve each problem step by step.
We are given polygons with some angles labeled, and we need to find the missing angle (marked as “x” or similar). To do this, we use the fact that:
> The sum of all interior angles in a polygon = (n – 2) × 180°, where n is the number of sides.
Then, subtract the known angles from the total to find the missing one.
---
Problem 1: Pentagon (5 sides)
Sum of interior angles = (5 – 2) × 180° = 3 × 180° = 540°
Given angles: 97°, 126°, 105°, 115°
Add them: 97 + 126 = 223; 223 + 105 = 328; 328 + 115 = 443°
Missing angle x = 540 – 443 = 97°
Wait — let me double-check addition:
97 + 126 = 223
223 + 105 = 328
328 + 115 = 443 → yes
540 – 443 = 97 → correct.
But wait — looking at the diagram again, there are FIVE angles shown? Let me count:
In Problem 1: Angles marked are: 97°, 126°, 105°, 115°, and x → that’s five angles. So if four are given and one is unknown, then yes, sum of four is 443, so x = 97°.
Actually, hold on — maybe I miscounted. Let me list them clearly:
From image description (assuming standard layout):
Problem 1: Pentagon with angles: 97°, 126°, 105°, 115°, and x → 5 angles total.
So sum of known four: 97 + 126 + 105 + 115
Calculate again:
97 + 126 = 223
105 + 115 = 220
223 + 220 = 443
540 – 443 = 97 → so x = 97°
Okay, seems right.
---
Problem 2: Hexagon (6 sides)
Sum = (6 – 2) × 180 = 4 × 180 = 720°
Given angles: 130°, 120°, 110°, 100°, 140°, and x → six angles.
Wait — actually, looking at typical problems like this, usually only five are given and one is missing. But here it says “find the measure of the indicated angle”, implying one is missing.
Assume five angles are given, one is x.
From image: angles are 130°, 120°, 110°, 100°, 140°, and x → that’s six values, but one is x.
So sum of known five: 130 + 120 + 110 + 100 + 140
Calculate:
130 + 120 = 250
110 + 100 = 210
250 + 210 = 460
460 + 140 = 600
Total needed: 720
So x = 720 – 600 = 120°
Check: 130+120=250; +110=360; +100=460; +140=600; 720-600=120 → correct.
---
Problem 3: Hexagon again?
Angles given: 115°, 120°, 110°, 105°, 125°, and x → six angles.
Sum for hexagon = 720°
Sum of known five: 115 + 120 + 110 + 105 + 125
Calculate:
115 + 120 = 235
110 + 105 = 215
235 + 215 = 450
450 + 125 = 575
x = 720 – 575 = 145°
Double-check: 115+120=235; +110=345; +105=450; +125=575; 720-575=145 → correct.
---
Problem 4: Pentagon
Sum = 540°
Angles: 100°, 110°, 120°, 105°, and x → five angles.
Sum of known four: 100 + 110 + 120 + 105
100 + 110 = 210
120 + 105 = 225
210 + 225 = 435
x = 540 – 435 = 105°
Check: 100+110=210; +120=330; +105=435; 540-435=105 → correct.
---
Problem 5: Heptagon? Wait — count sides.
Looking at shape: has 7 sides? Let's see — vertices: A,B,C,D,E,F,G → 7 sides → heptagon.
Sum = (7 – 2) × 180 = 5 × 180 = 900°
Given angles: 120°, 130°, 140°, 110°, 125°, 135°, and x → seven angles.
Sum of known six: 120 + 130 + 140 + 110 + 125 + 135
Calculate step by step:
120 + 130 = 250
140 + 110 = 250 → 250 + 250 = 500
125 + 135 = 260
500 + 260 = 760
x = 900 – 760 = 140°
Check: 120+130=250; +140=390; +110=500; +125=625; +135=760; 900-760=140 → correct.
---
Problem 6: Quadrilateral (4 sides)
Sum = (4 – 2) × 180 = 2 × 180 = 360°
Angles: 90°, 100°, 80°, and x → four angles.
Sum of known three: 90 + 100 + 80 = 270
x = 360 – 270 = 90°
Check: 90+100=190; +80=270; 360-270=90 → correct.
---
Now, compiling answers:
Problem 1: 97°
Problem 2: 120°
Problem 3: 145°
Problem 4: 105°
Problem 5: 140°
Problem 6: 90°
Final Answer:
1. 97°
2. 120°
3. 145°
4. 105°
5. 140°
6. 90°
We are given polygons with some angles labeled, and we need to find the missing angle (marked as “x” or similar). To do this, we use the fact that:
> The sum of all interior angles in a polygon = (n – 2) × 180°, where n is the number of sides.
Then, subtract the known angles from the total to find the missing one.
---
Problem 1: Pentagon (5 sides)
Sum of interior angles = (5 – 2) × 180° = 3 × 180° = 540°
Given angles: 97°, 126°, 105°, 115°
Add them: 97 + 126 = 223; 223 + 105 = 328; 328 + 115 = 443°
Missing angle x = 540 – 443 = 97°
Wait — let me double-check addition:
97 + 126 = 223
223 + 105 = 328
328 + 115 = 443 → yes
540 – 443 = 97 → correct.
But wait — looking at the diagram again, there are FIVE angles shown? Let me count:
In Problem 1: Angles marked are: 97°, 126°, 105°, 115°, and x → that’s five angles. So if four are given and one is unknown, then yes, sum of four is 443, so x = 97°.
Actually, hold on — maybe I miscounted. Let me list them clearly:
From image description (assuming standard layout):
Problem 1: Pentagon with angles: 97°, 126°, 105°, 115°, and x → 5 angles total.
So sum of known four: 97 + 126 + 105 + 115
Calculate again:
97 + 126 = 223
105 + 115 = 220
223 + 220 = 443
540 – 443 = 97 → so x = 97°
Okay, seems right.
---
Problem 2: Hexagon (6 sides)
Sum = (6 – 2) × 180 = 4 × 180 = 720°
Given angles: 130°, 120°, 110°, 100°, 140°, and x → six angles.
Wait — actually, looking at typical problems like this, usually only five are given and one is missing. But here it says “find the measure of the indicated angle”, implying one is missing.
Assume five angles are given, one is x.
From image: angles are 130°, 120°, 110°, 100°, 140°, and x → that’s six values, but one is x.
So sum of known five: 130 + 120 + 110 + 100 + 140
Calculate:
130 + 120 = 250
110 + 100 = 210
250 + 210 = 460
460 + 140 = 600
Total needed: 720
So x = 720 – 600 = 120°
Check: 130+120=250; +110=360; +100=460; +140=600; 720-600=120 → correct.
---
Problem 3: Hexagon again?
Angles given: 115°, 120°, 110°, 105°, 125°, and x → six angles.
Sum for hexagon = 720°
Sum of known five: 115 + 120 + 110 + 105 + 125
Calculate:
115 + 120 = 235
110 + 105 = 215
235 + 215 = 450
450 + 125 = 575
x = 720 – 575 = 145°
Double-check: 115+120=235; +110=345; +105=450; +125=575; 720-575=145 → correct.
---
Problem 4: Pentagon
Sum = 540°
Angles: 100°, 110°, 120°, 105°, and x → five angles.
Sum of known four: 100 + 110 + 120 + 105
100 + 110 = 210
120 + 105 = 225
210 + 225 = 435
x = 540 – 435 = 105°
Check: 100+110=210; +120=330; +105=435; 540-435=105 → correct.
---
Problem 5: Heptagon? Wait — count sides.
Looking at shape: has 7 sides? Let's see — vertices: A,B,C,D,E,F,G → 7 sides → heptagon.
Sum = (7 – 2) × 180 = 5 × 180 = 900°
Given angles: 120°, 130°, 140°, 110°, 125°, 135°, and x → seven angles.
Sum of known six: 120 + 130 + 140 + 110 + 125 + 135
Calculate step by step:
120 + 130 = 250
140 + 110 = 250 → 250 + 250 = 500
125 + 135 = 260
500 + 260 = 760
x = 900 – 760 = 140°
Check: 120+130=250; +140=390; +110=500; +125=625; +135=760; 900-760=140 → correct.
---
Problem 6: Quadrilateral (4 sides)
Sum = (4 – 2) × 180 = 2 × 180 = 360°
Angles: 90°, 100°, 80°, and x → four angles.
Sum of known three: 90 + 100 + 80 = 270
x = 360 – 270 = 90°
Check: 90+100=190; +80=270; 360-270=90 → correct.
---
Now, compiling answers:
Problem 1: 97°
Problem 2: 120°
Problem 3: 145°
Problem 4: 105°
Problem 5: 140°
Problem 6: 90°
Final Answer:
1. 97°
2. 120°
3. 145°
4. 105°
5. 140°
6. 90°
Parent Tip: Review the logic above to help your child master the concept of geometry worksheet polygon angle measures.
