Angles in Polygons worksheet - Free Printable

Let's solve each problem step by step using the formula for the sum of interior angles of a polygon:

> Sum of interior angles = $(n - 2) \times 180^\circ$,
> where $n$ is the number of sides.

We’ll apply this to each polygon and find the missing angle $x$.

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Problem 1: Quadrilateral (4 sides)

- Given angles: $60^\circ, 80^\circ, 120^\circ$, and $x$
- Number of sides $n = 4$

Step 1: Sum of interior angles
$$
(4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
$$

Step 2: Add known angles
$$
60^\circ + 80^\circ + 120^\circ = 260^\circ
$$

Step 3: Solve for $x$
$$
x = 360^\circ - 260^\circ = 100^\circ
$$

✔ Answer:
- Sum of interior angles = $360^\circ$
- $x = 100^\circ$

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Problem 2: Pentagon (5 sides)

- Given angles: $90^\circ, 122^\circ, 120^\circ, 103^\circ$, and $x$
- $n = 5$

Step 1: Sum of interior angles
$$
(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
$$

Step 2: Add known angles
$$
90 + 122 + 120 + 103 = 435^\circ
$$

Step 3: Solve for $x$
$$
x = 540^\circ - 435^\circ = 105^\circ
$$

✔ Answer:
- Sum of interior angles = $540^\circ$
- $x = 105^\circ$

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Problem 3: Hexagon (6 sides)

- Given angles: $108^\circ, 140^\circ, 128^\circ, 138^\circ, 130^\circ, 114^\circ$, and $x$
- Wait — count the angles: There are 6 angles shown, including $x$. So it's a hexagon with 6 sides → $n = 6$

Wait! Let’s double-check: The figure shows 6 angles, one of which is $x$, so total of 6 angles → yes, hexagon.

But wait — actually, looking at the diagram: It has 7 sides? Let's count the vertices. A hexagon has 6 sides and 6 angles.

But here we see 6 labeled angles, including $x$. So it must be a hexagon.

Wait — let’s list them:
- $x$, $108^\circ$, $140^\circ$, $128^\circ$, $138^\circ$, $130^\circ$, $114^\circ$ → that's 7 angles

Oops! That's 7 angles, so it's a heptagon (7 sides).

So $n = 7$

Step 1: Sum of interior angles
$$
(7 - 2) \times 180^\circ = 5 \times 180^\circ = 900^\circ
$$

Step 2: Add known angles
$$
108 + 140 + 128 + 138 + 130 + 114 = ?
$$
Break it down:
- $108 + 140 = 248$
- $248 + 128 = 376$
- $376 + 138 = 514$
- $514 + 130 = 644$
- $644 + 114 = 758^\circ$

Step 3: Solve for $x$
$$
x = 900^\circ - 758^\circ = 142^\circ
$$

✔ Answer:
- Sum of interior angles = $900^\circ$
- $x = 142^\circ$

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Problem 4: Heptagon (7 sides)

- Given angles: $120^\circ, 133^\circ, 110^\circ, 98^\circ, 148^\circ$, and $x$
- Count: 6 angles shown, one missing → total 7 angles → heptagon → $n = 7$

Step 1: Sum of interior angles
$$
(7 - 2) \times 180^\circ = 5 \times 180^\circ = 900^\circ
$$

Step 2: Add known angles
$$
120 + 133 + 110 + 98 + 148 = ?
$$
- $120 + 133 = 253$
- $253 + 110 = 363$
- $363 + 98 = 461$
- $461 + 148 = 609^\circ$

Step 3: Solve for $x$
$$
x = 900^\circ - 609^\circ = 291^\circ
$$

Wait! That seems too large — an interior angle of $291^\circ$? That would be reflex, but possible in concave polygons.

But let’s check if we missed something.

Wait — is this a heptagon? Let's count the sides from the image description.

In problem 4, the shape has 6 sides? No — the diagram shows a 6-sided figure? Wait — let's reconsider.

Actually, in problem 4, there are 6 angles shown: $120^\circ, 133^\circ, 110^\circ, 98^\circ, 148^\circ$, and $x$ → so 6 angles → it's a hexagon → $n = 6$

Ah! We made a mistake earlier.

Let’s re-analyze:

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Re-evaluate Problem 3 and 4

#### Problem 3:
Angles listed: $x, 108^\circ, 140^\circ, 128^\circ, 138^\circ, 130^\circ, 114^\circ$ → that’s 7 angles → so heptagon, $n = 7$ → correct.

Sum = $900^\circ$, known angles sum to $758^\circ$, so $x = 142^\circ$ ✔

#### Problem 4:
Angles: $120^\circ, 133^\circ, 110^\circ, 98^\circ, 148^\circ$, and $x$ → 6 angles → so it's a hexagon, $n = 6$

Sum of interior angles:
$$
(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ
$$

Add known angles:
$$
120 + 133 + 110 + 98 + 148 = ?
$$
- $120 + 133 = 253$
- $253 + 110 = 363$
- $363 + 98 = 461$
- $461 + 148 = 609^\circ$

Solve for $x$:
$$
x = 720^\circ - 609^\circ = 111^\circ
$$

✔ Answer:
- Sum of interior angles = $720^\circ$
- $x = 111^\circ$

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✔ Final Answers:

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1. Quadrilateral
- Sum of interior angles = $360^\circ$
- $x = 100^\circ$

2. Pentagon
- Sum of interior angles = $540^\circ$
- $x = 105^\circ$

3. Heptagon (7 sides)
- Sum of interior angles = $900^\circ$
- $x = 142^\circ$

4. Hexagon (6 sides)
- Sum of interior angles = $720^\circ$
- $x = 111^\circ$

---

Summary Table:

| Problem | Shape | Sum of Interior Angles | $x$ |
|--------|-------------|-------------------------|-----|
| 1 | Quadrilateral | $360^\circ$ | $100^\circ$ |
| 2 | Pentagon | $540^\circ$ | $105^\circ$ |
| 3 | Heptagon | $900^\circ$ | $142^\circ$ |
| 4 | Hexagon | $720^\circ$ | $111^\circ$ |

Let me know if you'd like a visual explanation or how to identify the number of sides!