Worksheet for finding missing angles in quadrilaterals, with problems and a solved example.

Let's solve each problem step by step using the key concept:

> The sum of the interior angles of any quadrilateral is 360°.

We will use this fact to find the missing angle(s) in each quadrilateral.

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Problem 1

Angles: 154°, 26°, 154°, and $ x^\circ $

$$
154 + 26 + 154 + x = 360 \\
334 + x = 360 \\
x = 360 - 334 = 26^\circ
$$

✔ Answer: $ x = 26^\circ $

---

Problem 2

Angles: $ x^\circ $, 82°, 77°, 88°

$$
x + 82 + 77 + 88 = 360 \\
x + 247 = 360 \\
x = 360 - 247 = 113^\circ
$$

✔ Answer: $ x = 113^\circ $

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Problem 3

Angles: 38°, 54°, 148°, $ x^\circ $

$$
38 + 54 + 148 + x = 360 \\
240 + x = 360 \\
x = 120^\circ
$$

✔ Answer: $ x = 120^\circ $

---

Problem 4

Angles: 120°, 66°, 60°, $ x^\circ $

$$
120 + 66 + 60 + x = 360 \\
246 + x = 360 \\
x = 114^\circ
$$

✔ Answer: $ x = 114^\circ $

---

Problem 5

Angles: $ x^\circ $, 150°, 100°, 66°

$$
x + 150 + 100 + 66 = 360 \\
x + 316 = 360 \\
x = 44^\circ
$$

✔ Answer: $ x = 44^\circ $

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Problem 6

This is a rectangle with two right angles (90°), one angle is 67°, and we need $ x^\circ $.

But wait — the figure shows two right angles (squares at corners), so two angles are 90° each.

Given:
- Two angles: 90°, 90°
- One angle: 67°
- One unknown: $ x^\circ $

$$
90 + 90 + 67 + x = 360 \\
247 + x = 360 \\
x = 113^\circ
$$

✔ Answer: $ x = 113^\circ $

> Note: This shape is not a rectangle because opposite angles aren't equal, but it's a quadrilateral with two right angles.

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Problem 7

This is a bit trickier. We have:
- Two right angles (90° each)
- One angle marked as $ y^\circ $
- One angle marked as $ x^\circ $
- And a circle around an angle labeled 348°, which is likely a reflex angle (greater than 180°).

But wait — a single interior angle can’t be 348° in a standard quadrilateral, because interior angles are less than 180° unless specified otherwise.

But here, 348° is outside the quadrilateral? Let’s look carefully.

Ah! The 348° is not an interior angle — it’s shown outside the corner, forming a full circle.

So, the interior angle at that vertex is:

$$
360^\circ - 348^\circ = 12^\circ
$$

Now, we know:
- Two angles are right angles: 90° each
- One interior angle is 12°
- One unknown: $ x^\circ $, and $ y^\circ $

Wait — actually, the diagram shows three angles:
- Two right angles (90°)
- One angle with reflex 348° → interior = 12°
- So only one angle is unknown?

But it asks for both $ x^\circ $ and $ y^\circ $. Looking closely, the labels suggest:

- $ x^\circ $ is at one corner (top-left)
- $ y^\circ $ is at bottom-left
- Top-right: 90°
- Bottom-right: reflex angle 348° → interior = 12°

So, let's list the four interior angles:
1. $ x^\circ $
2. 90° (top-right)
3. $ y^\circ $
4. 12° (bottom-right)

Sum = 360°

$$
x + 90 + y + 12 = 360 \\
x + y = 360 - 102 = 258^\circ
$$

But we need more info. Wait — are $ x $ and $ y $ adjacent? Is there symmetry?

Looking at the shape: It appears to be a rectangle with one corner "cut off", but the two right angles suggest it might be a trapezoid or irregular quadrilateral.

But notice: the top-left and bottom-left angles are labeled $ x $ and $ y $, and the top-right is 90°, bottom-right has reflex 348° → interior 12°.

But without more information, we cannot assume anything about $ x $ and $ y $ unless they are related.

Wait — perhaps the shape has two right angles, and the other two angles must add up to $ 360 - 90 - 90 - 12 = 168^\circ $? But no — we already used the 12°.

Wait — I think I made a mistake.

Let’s re-express:

The four interior angles are:
- Top-left: $ x^\circ $
- Top-right: 90°
- Bottom-right: interior angle = 360° - 348° = 12°
- Bottom-left: $ y^\circ $

So:
$$
x + 90 + 12 + y = 360 \\
x + y = 258^\circ
$$

But we still have two variables. Unless the shape implies something else...

Wait — maybe the angle labeled 348° is not the exterior angle but a typo or mislabeling?

Alternatively, perhaps the 348° is meant to represent a turn or exterior angle, but it's unusual.

But here's a better interpretation:

In some diagrams, when a large angle like 348° is shown outside the shape, it means that the interior angle is small.

So yes, interior angle at that vertex is $ 360° - 348° = 12° $

Now, the other three angles:
- Top-right: 90°
- Top-left: $ x $
- Bottom-left: $ y $

So total:
$$
x + 90 + y + 12 = 360 \Rightarrow x + y = 258^\circ
$$

But we need individual values.

Wait — perhaps the shape has two right angles, and the other two are $ x $ and $ y $, but unless it's symmetric, we can't determine them.

But looking again — the top-left and bottom-left are both labeled, but no additional info.

Wait — perhaps the 348° is not an exterior angle, but a mistake? Or maybe it's indicating that the sum of angles around a point?

No — it's drawn at a corner.

Alternatively, perhaps the 348° is a reflex interior angle, meaning the interior angle is 348°, but that would make the shape concave.

But if interior angle is 348°, then the other three angles must sum to $ 360 - 348 = 12^\circ $, which is impossible since each angle is positive and at least, say, 1°.

So interior angle cannot be 348° — too big.

Therefore, the 348° must be the reflex angle, so the interior angle is $ 360 - 348 = 12^\circ $

Now, back to the sum:
- Interior angles: $ x $, 90°, $ y $, 12°
- Sum: $ x + y + 102 = 360 \Rightarrow x + y = 258^\circ $

But we need more. Wait — the top-left and bottom-left angles may be adjacent, and if the side is straight, but no.

Unless the figure is symmetric?

Wait — look at the top-left and bottom-left — both are labeled $ x $ and $ y $, but not necessarily equal.

But the top-right is 90°, bottom-right has reflex 348° → interior 12°

So unless there’s another clue, we can't find individual values.

But wait — perhaps the angle labeled 348° is not part of the quadrilateral? That doesn’t make sense.

Alternatively, maybe the 348° is a typo — should be 348° as external?

But let's consider: in some problems, they show the reflex angle and ask for the interior.

But here, we are to find $ x $ and $ y $, so likely the shape has symmetry.

Wait — the top-left and bottom-left are both on the left side.

But unless it's a trapezoid or parallelogram, we can't assume.

But here’s a possibility: the left side is vertical, so the top-left and bottom-left angles might be supplementary? No — not necessarily.

Wait — the top-right is 90°, and the bottom-right has a reflex angle of 348°, so the interior is 12°.

So the sum of the other two angles is $ 360 - 90 - 12 = 258^\circ $

So $ x + y = 258^\circ $

But without more, we can’t find individual values.

But wait — the top-left and bottom-left are both labeled, and the shape looks like a rectangle with a very narrow cut, but no.

Perhaps the 348° is meant to be the sum of the angles around a point? But it's drawn at a corner.

Alternatively, maybe it's a mistake and should be 38° or something.

But let's look at the next problem.

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Problem 8

Quadrilateral with:
- Two angles: 58°, 58°
- One angle: 122°
- One unknown: $ x^\circ $

Sum:
$$
58 + 58 + 122 + x = 360 \\
238 + x = 360 \\
x = 122^\circ
$$

✔ Answer: $ x = 122^\circ $

Interesting — this is a kite-shaped quadrilateral, symmetric.

Now back to Problem 7.

Wait — perhaps the 348° is not an angle of the quadrilateral, but a label for the reflex angle outside, and the interior angle is 12°.

Then, the quadrilateral has:
- Top-right: 90°
- Bottom-right: 12°
- Top-left: $ x $
- Bottom-left: $ y $

And $ x + y = 258^\circ $

But we need more.

Wait — maybe the left side is a straight line? But no.

Another idea: perhaps the angle labeled 348° is not the reflex angle, but the total turn? Unlikely.

Alternatively, maybe the 348° is a typo and should be 38°?

But let's suppose it's correct.

Wait — maybe the quadrilateral has three right angles, and one reflex?

No — if three angles are 90°, sum is 270°, so fourth angle is 90° — can't be reflex.

But here, one angle is 12°, two are 90°, so third is 258° — impossible.

Wait — we have:
- $ x $, $ y $, 90°, 12° → sum 360° → $ x + y = 258^\circ $

But unless $ x $ and $ y $ are equal or related, we can't find them.

But look at the diagram: it shows two right angles at top-right and bottom-left? No — the right angles are at top-right and top-left?

Wait — the top-left is labeled $ x^\circ $, and there's a square symbol — so it's a right angle!

Ah! Look closely.

In Problem 7, the top-left corner has a square symbol — so it's a right angle.

Similarly, the top-right has a square symbol — so also 90°.

So:
- Top-left: 90° (because of square)
- Top-right: 90° (square)
- Bottom-right: reflex 348° → interior = 12°
- Bottom-left: $ y^\circ $

Now, we need to find $ x^\circ $ and $ y^\circ $

But $ x^\circ $ is at top-left — but there’s a square, so $ x = 90^\circ $

Similarly, $ y^\circ $ is at bottom-left — not marked with square.

So:
- $ x = 90^\circ $
- Top-right: 90°
- Bottom-right: interior = 12°
- Bottom-left: $ y $

Sum:
$$
90 + 90 + 12 + y = 360 \\
192 + y = 360 \\
y = 168^\circ
$$

✔ Answer: $ x = 90^\circ $, $ y = 168^\circ $

Yes! The square symbols indicate right angles.

So:
- $ x^\circ $ is at top-left — has square → $ x = 90^\circ $
- $ y^\circ $ is at bottom-left — unknown
- Top-right: 90°
- Bottom-right: reflex 348° → interior = 12°

So:
$$
90 + 90 + 12 + y = 360 \Rightarrow y = 168^\circ
$$

✔ Final Answer: $ x = 90^\circ $, $ y = 168^\circ $

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✔ Final Answers:

1. $ x = 26^\circ $
2. $ x = 113^\circ $
3. $ x = 120^\circ $
4. $ x = 114^\circ $
5. $ x = 44^\circ $
6. $ x = 113^\circ $
7. $ x = 90^\circ $, $ y = 168^\circ $
8. $ x = 122^\circ $

---

Summary Table:

| Problem | Answer |
|--------|--------|
| 1 | $ x = 26^\circ $ |
| 2 | $ x = 113^\circ $ |
| 3 | $ x = 120^\circ $ |
| 4 | $ x = 114^\circ $ |
| 5 | $ x = 44^\circ $ |
| 6 | $ x = 113^\circ $ |
| 7 | $ x = 90^\circ $, $ y = 168^\circ $ |
| 8 | $ x = 122^\circ $ |

Let me know if you'd like this formatted as a printable answer sheet!