This worksheet guides students through calculating the sum of interior angles and solving for missing variables in irregular polygons like pentagons and hexagons.

Let’s solve each problem step by step.

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Problem 1: Quadrilateral PQRS

This is a 4-sided polygon (quadrilateral).

→ Sum of interior angles = (Number of sides – 2) × 180°
= (4 – 2) × 180° = 2 × 180° = 360°

Now, add up all the given angles and set equal to 360°:

Angle P = x
Angle Q = 105°
Angle R = x – 6°
Angle S = x + 12°
Angle T? Wait — hold on! The diagram shows points P, Q, R, S, T — that’s 5 points. But it's drawn as a quadrilateral? Let me check again.

Looking at the labels: P, Q, R, S, T — that’s 5 vertices → pentagon!

Wait — in the diagram for #1, there are 5 labeled corners: P, Q, R, S, T. So it’s a pentagon, not a quadrilateral.

So number of sides = 5

→ Sum of interior angles = (5 – 2) × 180° = 3 × 180° = 540°

Now write equation:

x (P) + 105° (Q) + (x – 6°) (R) + (x + 12°) (S) + 75° (T) = 540°

Combine like terms:

x + x + x = 3x
Constants: 105 – 6 + 12 + 75 = let’s compute:

105 + 75 = 180
–6 + 12 = +6
Total constants = 180 + 6 = 186

So:
3x + 186 = 540
Subtract 186 from both sides:
3x = 540 – 186 = 354
Divide by 3:
x = 354 ÷ 3 = 118

Now find each angle:

∠P = x = 118°
∠R = x – 6 = 118 – 6 = 112°
∠S = x + 12 = 118 + 12 = 130°

Check sum:
118 (P) + 105 (Q) + 112 (R) + 130 (S) + 75 (T) =
118 + 105 = 223
223 + 112 = 335
335 + 130 = 465
465 + 75 = 540 ✔ Correct.

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Problem 2: Heptagon ABCDEFG

Count the vertices: A, B, C, D, E, F, G → 7 sides → heptagon

Sum of interior angles = (7 – 2) × 180° = 5 × 180° = 900°

Given angles:

A: x + 18°
B: x
C: 138°
D: x + 10°
E: 130°
F: 130°
G: 132°

Add them all:

(x + 18) + x + 138 + (x + 10) + 130 + 130 + 132

Combine x terms: x + x + x = 3x

Constants: 18 + 138 + 10 + 130 + 130 + 132

Compute step by step:

18 + 138 = 156
156 + 10 = 166
166 + 130 = 296
296 + 130 = 426
426 + 132 = 558

So total: 3x + 558 = 900

Subtract 558:
3x = 900 – 558 = 342

Divide by 3:
x = 342 ÷ 3 = 114

Now find required angles:

∠A = x + 18 = 114 + 18 = 132°
∠B = x = 114°
∠D = x + 10 = 114 + 10 = 124°

Check sum:
A:132, B:114, C:138, D:124, E:130, F:130, G:132

Add:
132 + 114 = 246
246 + 138 = 384
384 + 124 = 508
508 + 130 = 638
638 + 130 = 768
768 + 132 = 900 ✔ Correct.

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Problem 3: Hexagon GHIJKL

Vertices: G, H, I, J, K, L → 6 sides → hexagon

Sum of interior angles = (6 – 2) × 180° = 4 × 180° = 720°

Given angles:

G: x – 27°
H: right angle → 90° (marked with square)
I: x + 30°
J: 112°
K: 125°
L: x + 15°

Add them:

(x – 27) + 90 + (x + 30) + 112 + 125 + (x + 15)

x terms: x + x + x = 3x

Constants: –27 + 90 + 30 + 112 + 125 + 15

Compute step by step:

Start with –27 + 90 = 63
63 + 30 = 93
93 + 112 = 205
205 + 125 = 330
330 + 15 = 345

So: 3x + 345 = 720

Subtract 345:
3x = 720 – 345 = 375

Divide by 3:
x = 375 ÷ 3 = 125

Now find required angles:

∠G = x – 27 = 125 – 27 = 98°
∠I = x + 30 = 125 + 30 = 155°
∠L = x + 15 = 125 + 15 = 140°

Check sum:
G:98, H:90, I:155, J:112, K:125, L:140

Add:
98 + 90 = 188
188 + 155 = 343
343 + 112 = 455
455 + 125 = 580
580 + 140 = 720 ✔ Correct.

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Final Answer:

1) Sum of the interior angles = 540°
x = 118; ∠P = 118°; ∠R = 112°; ∠S = 130°

2) Sum of the interior angles = 900°
x = 114; ∠A = 132°; ∠B = 114°; ∠D = 124°

3) Sum of the interior angles = 720°
x = 125; ∠G = 98°; ∠I = 155°; ∠L = 140°