Math worksheet for finding interior and exterior angles of polygons, featuring various geometric shapes with labeled angles and algebraic expressions.

Let’s solve each polygon one by one. We’ll use the fact that:

- For any polygon, the sum of interior angles = (n – 2) × 180°, where n is the number of sides.
- An exterior angle and its adjacent interior angle add up to 180°.
- If an exterior angle is given, we can find the interior angle by subtracting from 180°, and vice versa.

We’ll go shape by shape.

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Shape 1: Pentagon (5 sides)

Interior angles:
x + 75°, x, 2x, 140°, 33°

Sum of interior angles for pentagon = (5–2)×180 = 540°

So:

(x + 75) + x + 2x + 140 + 33 = 540
Combine like terms:
x + x + 2x = 4x
75 + 140 + 33 = 248

So:
4x + 248 = 540
4x = 540 – 248 = 292
x = 292 ÷ 4 = 73

✔ Check:
Angles: 73+75=148, 73, 2×73=146, 140, 33 → 148+73=221; 221+146=367; 367+140=507; 507+33=540 ✔️

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Shape 2: Pentagon with some exterior angles

Given:
- Interior angles: x°, 110°, ? , ? , ?
- Exterior angles: 80°, 42°, and two others are labeled as part of expressions? Wait — let’s look again.

Actually, looking at the diagram:

It shows a pentagon with:
- One interior angle: x°
- Another interior: 110°
- Two exterior angles shown: 80° and 42° → so their adjacent interior angles are 180–80=100°, and 180–42=138°
- Also, there's an interior angle labeled “2x°”

Wait — actually, in the diagram, it looks like:

The five interior angles are:
- x°
- 110°
- 2x°
- and two more: one adjacent to 80° exterior → so interior = 100°
- one adjacent to 42° exterior → so interior = 138°

So total interior angles: x, 110, 2x, 100, 138

Sum = 540°

So:

x + 110 + 2x + 100 + 138 = 540
3x + 348 = 540
3x = 540 – 348 = 192
x = 192 ÷ 3 = 64

✔ Check:
Angles: 64, 110, 128 (2×64), 100, 138 → 64+110=174; +128=302; +100=402; +138=540 ✔️

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Shape 3: Pentagon

Interior angles: 4x, x, 2x+9, 3x+8, 113°

Sum = 540°

So:

4x + x + (2x+9) + (3x+8) + 113 = 540
Combine x terms: 4x+x+2x+3x = 10x
Constants: 9+8+113 = 130

So:
10x + 130 = 540
10x = 410
x = 41

✔ Check:
4×41=164, 41, 2×41+9=91, 3×41+8=131, 113
Sum: 164+41=205; +91=296; +131=427; +113=540 ✔️

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Shape 4: Quadrilateral (4 sides)

Interior angles: 84°, x°, 102°, and one exterior angle 122° → so adjacent interior = 180–122=58°

Sum of interior angles for quadrilateral = (4–2)×180 = 360°

So:

84 + x + 102 + 58 = 360
x + 244 = 360
x = 360 – 244 = 116

✔ Check: 84+116=200; +102=302; +58=360 ✔️

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Shape 5: Quadrilateral with right angle and exterior angles

Diagram shows:
- Right angle (90°)
- Exterior angle 68° → interior = 180–68=112°
- Exterior angle 117° → interior = 180–117=63°
- One interior angle labeled 2y

Sum = 360°

So:

90 + 112 + 63 + 2y = 360
265 + 2y = 360
2y = 95
y = 47.5

✔ Check: 90+112=202; +63=265; +95=360 ✔️

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Shape 6: Hexagon (6 sides)

Exterior angles given: 46°, 93°, 47°, 62°, and two others: 3x° and 4x°

Important: Sum of exterior angles of ANY polygon = 360° (always!)

So:

46 + 93 + 47 + 62 + 3x + 4x = 360
Add constants: 46+93=139; +47=186; +62=248
So: 248 + 7x = 360
7x = 360 – 248 = 112
x = 112 ÷ 7 = 16

✔ Check: 3x=48, 4x=64 → 46+93+47+62+48+64 = let’s add: 46+93=139; +47=186; +62=248; +48=296; +64=360 ✔️

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Shape 7: Pentagon with mixed interior/exterior

Diagram shows:
- Right angle (interior 90°)
- Exterior angle 70° → interior = 110°
- Exterior angle 60° → interior = 120°
- Another right angle? Wait — there’s a 90° marked on the outside? Let’s interpret carefully.

Looking at the figure:

There are 5 vertices.

At bottom left: exterior angle 70° → interior = 110°
At bottom right: interior angle is 90° (marked with square)
At top right: exterior angle 60° → interior = 120°
At top left: interior angle is 90° (right angle symbol inside)
And one unknown interior angle: x°

Wait — but if both top-left and bottom-right are 90°, and we have interiors: 110°, 90°, 120°, 90°, and x°

Sum should be 540°

So:

110 + 90 + 120 + 90 + x = 540
410 + x = 540
x = 130

But wait — let me double-check the diagram interpretation.

Actually, in the image, it might be that the 90° at bottom right is an *exterior* angle? No — it’s drawn with the square symbol *inside* the corner, so it’s interior.

Similarly, top left has square symbol inside → interior 90°.

Bottom left: line extended, angle between extension and side is 70° → that’s exterior → interior = 110°

Top right: line extended, angle between extension and side is 60° → exterior → interior = 120°

Then the fifth angle is x° — which appears to be at the bottom middle? Or wherever.

Yes, so interiors: 110°, 90°, 120°, 90°, x° → sum 540°

So x = 540 - (110+90+120+90) = 540 - 410 = 130

✔ Confirmed.

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Shape 8: Trapezoid (quadrilateral)

Interior angles: 85°, 75°, x°, and one exterior angle 108° → so adjacent interior = 180–108=72°

Sum = 360°

So:

85 + 75 + x + 72 = 360
232 + x = 360
x = 360 – 232 = 128

✔ Check: 85+75=160; +72=232; +128=360 ✔️

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Final Answers:

Shape 1: x = 73
Shape 2: x = 64
Shape 3: x = 41
Shape 4: x = 116
Shape 5: y = 47.5
Shape 6: x = 16
Shape 7: x = 130
Shape 8: x = 128

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Final Answer:
Shape 1: x = 73
Shape 2: x = 64
Shape 3: x = 41
Shape 4: x = 116
Shape 5: y = 47.5
Shape 6: x = 16
Shape 7: x = 130
Shape 8: x = 128