Find the value of "x" in each triangle using angle sum properties and algebra.

Let's solve each of these triangle angle problems step by step. The key concept we'll use is:

> The sum of the interior angles in any triangle is always 180°.

We’ll go through each problem one at a time.

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1)

Triangle with angles: 58°, 47°, and $ x^\circ $

$$
x = 180^\circ - 58^\circ - 47^\circ = 180 - 105 = 75^\circ
$$

✔ Answer: $ x = 75 $

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2)

Angles: $ x^\circ $, 96°, 21°

$$
x = 180 - 96 - 21 = 180 - 117 = 63^\circ
$$

✔ Answer: $ x = 63 $

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3)

Right triangle (has a right angle), so one angle is 90°, another is 31°, and the third is $ (3x - 1)^\circ $

Sum of angles:
$$
90 + 31 + (3x - 1) = 180 \\
120 + 3x - 1 = 180 \\
119 + 3x = 180 \\
3x = 61 \\
x = \frac{61}{3} \approx 20.33
$$

Wait — let’s double-check.

Actually, the angle labeled $ (3x - 1)^\circ $ is likely one of the non-right angles, and the other is 31°.

So:
$$
90 + 31 + (3x - 1) = 180 \\
120 + 3x - 1 = 180 \\
119 + 3x = 180 \\
3x = 61 \\
x = \frac{61}{3} \approx 20.33
$$

But this gives a fractional value. Let’s check if that makes sense.

Wait — perhaps the labeling is different? Let’s assume the triangle has a right angle (90°), one angle is 31°, and the third is $ (3x - 1)^\circ $. Then yes, it must be:

$$
(3x - 1) = 180 - 90 - 31 = 59^\circ \\
3x - 1 = 59 \\
3x = 60 \\
x = 20
$$

Ah! So $ x = 20 $, and $ 3x - 1 = 59^\circ $, which works.

✔ Answer: $ x = 20 $

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4)

Angles: $ x^\circ $, 21°, 34°

$$
x = 180 - 21 - 34 = 180 - 55 = 125^\circ
$$

✔ Answer: $ x = 125 $

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5)

Angles: $ (2x + 3)^\circ $, 100°, 51°

$$
(2x + 3) + 100 + 51 = 180 \\
2x + 154 = 180 \\
2x = 26 \\
x = 13
$$

✔ Answer: $ x = 13 $

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6)

Angles: $ x^\circ $, 60°, $ 2x^\circ $, and the third angle is 94°?

Wait — triangle has three angles. One is 94°, one is 60°, and the third is $ 2x^\circ $? But label says $ x^\circ $ and $ 2x^\circ $? Wait — look again.

It shows: one angle is 94°, one is 60°, and the third is $ 2x^\circ $. So:

$$
94 + 60 + 2x = 180 \\
154 + 2x = 180 \\
2x = 26 \\
x = 13
$$

✔ Answer: $ x = 13 $

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7)

Angles: 52°, 43°, $ x^\circ $

$$
x = 180 - 52 - 43 = 180 - 95 = 85^\circ
$$

✔ Answer: $ x = 85 $

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8)

Angles: $ x^\circ $, $ 3x^\circ $, $ 2x^\circ $

All three angles are expressed in terms of $ x $:

$$
x + 3x + 2x = 180 \\
6x = 180 \\
x = 30
$$

✔ Answer: $ x = 30 $

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9)

Right triangle (right angle shown), so one angle is 90°.

Other angles: $ (2x - 2)^\circ $, and $ (x + 5)^\circ $

So:
$$
90 + (2x - 2) + (x + 5) = 180 \\
90 + 2x - 2 + x + 5 = 180 \\
(90 - 2 + 5) + 3x = 180 \\
93 + 3x = 180 \\
3x = 87 \\
x = 29
$$

Check:
- $ 2x - 2 = 58 - 2 = 56^\circ $
- $ x + 5 = 34^\circ $
- $ 90 + 56 + 34 = 180 $ ✔

✔ Answer: $ x = 29 $

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10)

This is not a triangle — it looks like an angle formed by two lines extending from a point, but it's not a triangle. It appears to be a straight line or an external angle.

Wait — it shows two angles: 10° and 23°, and then $ x^\circ $ as the angle between them?

Wait — actually, it looks like a triangle with an exterior angle? Or maybe a straight line?

Looking closely: There’s a triangle-like shape, but it seems like three angles meeting at a point — no, more likely it's a triangle with two known angles on the base, and $ x $ is the top angle.

Wait — the diagram shows a triangle with two base angles: 10° and 23°, and $ x^\circ $ as the vertex angle?

Then:

$$
x = 180 - 10 - 23 = 147^\circ
$$

But wait — that would make the triangle very "flat", but mathematically possible.

Alternatively, could it be an exterior angle?

But based on standard interpretation: triangle with two angles given: 10° and 23°, and $ x $ is the third angle.

So:

$$
x = 180 - 10 - 23 = 147^\circ
$$

✔ Answer: $ x = 147 $

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11)

Equilateral triangle? All angles are $ x^\circ $, and there are three angles.

So:
$$
x + x + x = 180 \\
3x = 180 \\
x = 60
$$

✔ Answer: $ x = 60 $

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12)

Angles: $ (3x - 17)^\circ $, $ (x + 40)^\circ $, $ (2x - 5)^\circ $

Sum to 180°:

$$
(3x - 17) + (x + 40) + (2x - 5) = 180 \\
3x - 17 + x + 40 + 2x - 5 = 180 \\
(3x + x + 2x) + (-17 + 40 - 5) = 180 \\
6x + 18 = 180 \\
6x = 162 \\
x = 27
$$

Now verify:
- $ 3x - 17 = 81 - 17 = 64^\circ $
- $ x + 40 = 27 + 40 = 67^\circ $
- $ 2x - 5 = 54 - 5 = 49^\circ $
- Sum: $ 64 + 67 + 49 = 180 $ ✔

✔ Answer: $ x = 27 $

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✔ Final Answers:

1) $ x = 75 $
2) $ x = 63 $
3) $ x = 20 $
4) $ x = 125 $
5) $ x = 13 $
6) $ x = 13 $
7) $ x = 85 $
8) $ x = 30 $
9) $ x = 29 $
10) $ x = 147 $
11) $ x = 60 $
12) $ x = 27 $

Let me know if you'd like a visual explanation or help with any specific one!