Practice worksheet for calculating missing angles in triangles.

Let's solve the problems from the image step by step. The worksheet is titled "Angles in a Triangle", and it contains two main parts:

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Part 1: Find the missing angle marked $ x $ in each triangle

We use the triangle angle sum theorem:
> The sum of the interior angles in any triangle is always 180°.

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#### A.
Given angles:
- One angle = $31^\circ$
- Another angle = $48^\circ$

So:
$$
x = 180^\circ - 31^\circ - 48^\circ = 101^\circ
$$

✔ Answer: $ x = 101^\circ $

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#### B.
Given angles:
- One angle = $57^\circ$
- Another angle = $64^\circ$

So:
$$
x = 180^\circ - 57^\circ - 64^\circ = 59^\circ
$$

✔ Answer: $ x = 59^\circ $

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#### C.
Given angles:
- One angle = $9^\circ$
- Another angle = $37^\circ$

So:
$$
x = 180^\circ - 9^\circ - 37^\circ = 134^\circ
$$

✔ Answer: $ x = 134^\circ $

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#### D.
This triangle has an exterior angle marked as $110^\circ$, and one interior angle is $42^\circ$. We are to find $ x $, which appears to be the third interior angle.

But note: The exterior angle at a vertex is equal to the sum of the two opposite interior angles.

So:
$$
\text{Exterior angle} = \text{sum of two remote interior angles}
$$
$$
110^\circ = 42^\circ + x
$$
$$
x = 110^\circ - 42^\circ = 68^\circ
$$

✔ Answer: $ x = 68^\circ $

Alternatively, you could find the interior angle adjacent to the $110^\circ$ exterior angle:
$$
180^\circ - 110^\circ = 70^\circ
$$
Then use triangle sum:
$$
x = 180^\circ - 42^\circ - 70^\circ = 68^\circ
$$

Same result.

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✔ Summary for Part 1:

| Problem | Answer |
|--------|--------|
| A | $101^\circ$ |
| B | $59^\circ$ |
| C | $134^\circ$ |
| D | $68^\circ$ |

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Part 2: Find the size of the angle marked $ x $

This diagram shows a triangle with two known angles: $80^\circ$ and $34^\circ$, and a line drawn from the top vertex to the base, forming two smaller triangles. But the angle $ x $ is labeled at the bottom right corner.

Wait — let’s analyze carefully.

From the diagram (based on typical layout):

- A large triangle with angles $80^\circ$ at the top, $34^\circ$ at the left bottom.
- A line is drawn from the top vertex to a point on the base, creating two smaller triangles.
- Angle $ x $ is at the bottom right of the original triangle.

But we don’t have enough info unless we assume that the line splits the triangle into two parts, but no other angles are given.

Wait — perhaps the angle $ x $ is just the third angle of the triangle?

Let’s suppose that the triangle has:
- Top angle: $80^\circ$
- Left bottom angle: $34^\circ$
- Right bottom angle: $ x $

Then:
$$
x = 180^\circ - 80^\circ - 34^\circ = 66^\circ
$$

✔ So $ x = 66^\circ $

But wait — there’s a line drawn from the top vertex to the base, so maybe $ x $ is not the full bottom angle? Let’s recheck.

Actually, looking at the diagram description:
It says “Find the size of the angle marked $ x $” — and $ x $ is at the bottom right corner, likely part of the original triangle.

But if the line splits the triangle, and $ x $ is one of the angles formed, we need more data.

However, since only two angles ($80^\circ$ and $34^\circ$) are given, and $ x $ is the third angle, then yes:

$$
x = 180^\circ - 80^\circ - 34^\circ = 66^\circ
$$

✔ Answer: $ x = 66^\circ $

Even if there’s a line drawn, unless it's specifying another angle, and since $ x $ is clearly labeled at the bottom right vertex, it must be the angle at that corner.

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✔ Final Answers:

#### Part 1:
- A: $ \boxed{101^\circ} $
- B: $ \boxed{59^\circ} $
- C: $ \boxed{134^\circ} $
- D: $ \boxed{68^\circ} $

#### Part 2:
- $ \boxed{66^\circ} $

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Let me know if you'd like a visual explanation or clarification on any step!