Answer Key - Unit 8 Test Study Guide (Polygons & Quadrilaterals ... - Free Printable

Let’s go through each problem step by step to verify the answers shown in the study guide. We’ll check calculations and logic for accuracy.

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Problem 1: Formula for sum of interior angles of a polygon?
Answer given: S = (n - 2) × 180
✔ Correct. This is the standard formula.

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Problem 2: Formula for each interior angle of a regular polygon?
Answer given: [(n - 2) × 180] / n
✔ Correct. Divide total sum by number of sides.

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Problem 3: Sum of interior angles of a 35-gon?
Calculation: (35 - 2) × 180 = 33 × 180
33 × 180 = 33 × 100 + 33 × 80 = 3300 + 2640 = 5940°
✔ Answer correct.

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Problem 4: Heptagon with six known angles — find seventh?
Heptagon has 7 sides → sum = (7-2)×180 = 900°
Given angles: 107, 139, 131, 110, 145, 128
Add them:
107 + 139 = 246
246 + 131 = 377
377 + 110 = 487
487 + 145 = 632
632 + 128 = 760°
Then: 900 - 760 = 140°
✔ Answer correct.

*(Note: The student wrote “900 - 760 = 140” but mistakenly wrote “760” as “760” — wait, they actually wrote “900 - 760 = 140”, which is right. But in their work they said “900 - 760 = 140” — yes, that’s fine. Actually looking again: they wrote “900 - 760 = 140” — correct. Wait, no — in the image it says “900 - 760 = 140”? Let me recheck: In the image, under #4, they have:

(7-2)*180 = 900
then 900 - 760 = 140 — but where did 760 come from? They must have added the six angles correctly to get 760. Let’s add again:

107 + 139 = 246
+131 = 377
+110 = 487
+145 = 632
+128 = 760 → yes. So 900 - 760 = 140. ✔ Correct.

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Problem 5: If sum of interior angles is 3780°, how many sides?
Equation: (n - 2) × 180 = 3780
Divide both sides by 180:
3780 ÷ 180 = ?
180 × 20 = 3600
3780 - 3600 = 180 → so 21
So n - 2 = 21 → n = 23
✔ Answer correct.

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Problem 6: Each interior angle of regular 18-gon?
Sum = (18-2)×180 = 16×180 = 2880
Each angle = 2880 ÷ 18
2880 ÷ 18:
18 × 160 = 2880 → yes, 160°
✔ Answer correct.

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Problem 7: Sum of exterior angles of any polygon?
Always 360° — regardless of sides.
✔ Answer correct.

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Problem 8: Each exterior angle of regular 30-gon?
360 ÷ 30 = 12°
✔ Answer correct.

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Problem 9: Exterior angle = 24° → how many sides?
For regular polygon: 360 ÷ exterior angle = number of sides
360 ÷ 24 = 15
✔ Answer correct.

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Problem 10: Interior angle = 162° → how many sides?
Exterior angle = 180 - 162 = 18°
Number of sides = 360 18 = 20
✔ Answer correct.

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Problem 11: Solve for x in octagon with given angles.

Octagon → 8 sides → sum = (8-2)×180 = 1080°

Angles given:
139°, (10x - 11)°, 121°, (9x - 17)°, (7x + 6)°, 153°, (9x - 14)°, (8x + 15)°

Add all constants first:
139 + 121 = 260
260 + 153 = 413
Now variables:
10x + 9x + 7x + 9x + 8x = 43x
Constants from variable terms:
-11 -17 +6 -14 +15 = let's compute:
-11 -17 = -28
-28 +6 = -22
-22 -14 = -36
-36 +15 = -21

Wait — we also have the fixed numbers: 139, 121, 153 → already added to 413

But the other three angles are expressed with x: (10x-11), (9x-17), (7x+6), (9x-14), (8x+15) — that’s five expressions with x, plus three constant angles: 139, 121, 153.

Total angles: 8.

List again:

1. 139
2. 10x - 11
3. 121
4. 9x - 17
5. 7x + 6
6. 153
7. 9x - 14
8. 8x + 15

So constants: 139 + 121 + 153 = 413
Plus constants from expressions: -11 -17 +6 -14 +15 = (-11-17= -28; -28+6=-22; -22-14=-36; -36+15= -21)

Total constants: 413 + (-21) = 392

Variables: 10x + 9x + 7x + 9x + 8x = 43x

So equation: 43x + 392 = 1080
43x = 1080 - 392 = 688
x = 688 ÷ 43

Check: 43 × 16 = 688 → yes!
✔ x = 16 — correct.

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Problem 12: Regular pentagon → solve for x if one angle is (7x + 31)°

Pentagon → 5 sides → each interior angle = (5-2)×180 / 5 = 540 / 5 = 108°

Set equal: 7x + 31 = 108
7x = 77
x = 11
✔ Correct.

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Problem 13: Polygon with exterior angles given — solve for x.

Figure shows a hexagon? Wait — count the arrows/exterior angles: there are 6 exterior angles shown.

Labels:
(11x - 5)°
(4x + 29)°
61°
(6x + 13)°
(9x - 15)°
67°

Sum of exterior angles of ANY polygon = 360°

So add all:

(11x - 5) + (4x + 29) + 61 + (6x + 13) + (9x - 15) + 67

Combine x terms: 11x + 4x + 6x + 9x = 30x

Constants: -5 + 29 + 61 + 13 -15 + 67
Compute step by step:
-5 + 29 = 24
24 + 61 = 85
85 + 13 = 98
98 - 15 = 83
83 + 67 = 150

So: 30x + 150 = 360
30x = 210
x = 7
✔ Correct.

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All answers in the study guide are mathematically correct.

Final Answer: All problems solved correctly as shown.