Special Angles - Free Printable

It looks like you've uploaded an image containing several geometry problems involving special angles and angle relationships in triangles and coordinate planes. Let's go through each problem one by one, solve them, and explain the reasoning.

---

Problem 1: Triangle with Angles

```
Triangle ABC:
- Angle B = 35°
- Angle C = 90° (right angle)
- Find angle A
```

#### Given:
- ∠B = 35°
- ∠C = 90°

We know that the sum of angles in a triangle is 180°.

So,
$$
∠A + ∠B + ∠C = 180°
$$
$$
∠A + 35° + 90° = 180°
$$
$$
∠A = 180° - 125° = 55°
$$

✔ So, ∠A = 55°

But in the image, it says:
> B = 35°
> B + 35° = 180° → This seems incorrect.

Wait — there’s a typo or confusion here. It says:
> B = 35°
> B + 35° = 180° → which would imply 35 + 35 = 180? That’s wrong.

Let’s look again:

Actually, it appears to say:
> B = 35°
> B + 35° = 180°
> -35
> B = 145° ← this doesn't make sense.

This part has an error in logic.

But from the diagram:
- Right triangle at C → ∠C = 90°
- ∠B = 35°
- So ∠A = 180 - 90 - 35 = 55°

✔ Correct answer: ∠A = 55°, not 145°.

The red annotation appears to be incorrect.

---

Problem 2: Triangle with Variable Angle

```
Triangle ABC:
- ∠B = 72°
- ∠A = 2x - 6
- ∠C = ?
- Sum of angles = 180°
```

But wait — the triangle shows two angles labeled:
- ∠B = 72°
- ∠A = 2x - 6
- And we’re told that ∠A + ∠B = 180°? That can’t be right unless it's a straight line.

Wait — looking at the diagram:
- Points A, B, C form a triangle.
- But the notation shows arrows at A and B, suggesting parallel lines?

Wait — actually, this might be a straight line with angles around point C.

Let me re-analyze:

From the image:
- There are two rays from point C: CA and CB
- ∠ACB = 72°
- ∠A = 2x - 6
- But the equation written is:
> 2x - 6 = 72°
> 2x = 78
> x = 39°

So they're saying: 2x - 6 = 72, so x = 39.

But why?

Possibly, the figure is showing two angles forming a straight line? Or maybe it's a triangle where two angles are related.

Wait — perhaps the triangle has:
- ∠A = 2x - 6
- ∠B = 72°
- And the third angle is unknown?

But then they write:
> 2x - 6 = 72 → x = 39

That implies ∠A = 72°, so if ∠A = ∠B, then it's isosceles?

But unless stated, that assumption isn't valid.

Alternatively, could this be a linear pair?

Looking at the diagram: Point C, with rays going to A and B, and angles labeled.

Wait — actually, the correct interpretation may be:

There is a triangle ABC with:
- ∠B = 72°
- ∠A = 2x - 6
- And possibly ∠C is given or implied?

But no — the work says:
> 2x - 6 = 72
> 2x = 78
> x = 39

So they assume ∠A = 72°, which would mean 2x - 6 = 72 → x = 39.

But why?

Unless the triangle has two equal angles, or it's a special case.

Alternatively, could this be a straight line?

Wait — the diagram shows vectors from C to A and C to B, but no triangle.

Ah! Perhaps it's a straight angle at point C, and the two angles add up to 180°?

But only one angle is labeled: 72°, and the other is 2x - 6.

Then:
> (2x - 6) + 72 = 180
> 2x + 66 = 180
> 2x = 114
> x = 57

But the image says:
> 2x - 6 = 72 → x = 39

That would make the angle 72°, so both angles are 72°? Then total = 144°, not 180.

So contradiction.

Wait — maybe it's a triangle with:
- ∠A = 2x - 6
- ∠B = 72°
- ∠C = ? (not shown)

But without more info, we can't solve.

But the image says:
> 2x - 6 = 72 → x = 39

So perhaps it's assumed that ∠A = ∠B = 72°, so 2x - 6 = 72 → x = 39

Then ∠A = 72°, ∠B = 72°, so ∠C = 180 - 72 - 72 = 36°

So it's an isosceles triangle.

But unless stated, we can't assume equality.

However, the way the equation is written suggests that the angle labeled as "2x - 6" is equal to 72°.

So likely, 2x - 6 = 72 is given as an equation.

Then:
> 2x = 78
> x = 39

✔ So x = 39°

Even though the logic is unclear, the math is correct if 2x - 6 = 72.

So x = 39

---

Problem 3: Right Triangle with Angle x

```
Right triangle at C
- One angle = 2x
- Other angle = x
- Right angle = 90°
```

From the diagram:
- Right angle at C
- ∠A = 2x
- ∠B = x

Sum of angles:
> 2x + x + 90° = 180°
> 3x = 90°
> x = 30°

Then:
- ∠A = 2(30) = 60°
- ∠B = 30°

✔ So x = 30°, and angles are 30°, 60°, 90° — classic special triangle.

The solution in the image is correct:
> x + 2x = 90°
> 3x = 90
> x = 30
> 2x = 60°

Perfect.

---

Problem 4: Perpendicular Lines and Angle WOX

```
Point O on line WZ
- Ray OX forms 40° with OY
- OY ⊥ WZ → so ∠WOY = 90°
- Find m∠WOX
```

From diagram:
- WZ is a straight line
- OY is perpendicular to WZ → so ∠WOY = 90°
- ∠YOX = 40°
- We want ∠WOX

Since ∠WOX = ∠WOY + ∠YOX = 90° + 40° = 130°

But the image says:
> m∠WOX + 40° = 90° → this is wrong!

Wait — that would imply ∠WOX = 50°

But that’s incorrect.

Let’s see:
- If OY is perpendicular to WZ, then ∠WOY = 90°
- OX is above OY, making 40° with OY
- So ∠WOX = ∠WOY + ∠YOX = 90° + 40° = 130°

But the image says:
> m∠WOX + 40° = 90°
> -40
> m∠WOX = 50°

✘ That’s incorrect.

Why would someone think that?

Only if OX were between W and Y, but the diagram shows OX going upward from O, and OY is vertical, so OX is 40° from OY, meaning it's outside the 90° angle.

So ∠WOX = 90° + 40° = 130°

✔ Correct answer: m∠WOX = 130°

But the image says 50°, which is wrong.

---

✔ Summary of Solutions:

| Problem | Correct Answer | Explanation |
|--------|----------------|-----------|
| 1 | ∠A = 55° | In a right triangle, 90° + 35° = 125°, so remaining angle = 55° |
| 2 | x = 39° | Given 2x - 6 = 72 → 2x = 78 → x = 39 |
| 3 | x = 30° | 2x + x = 90° → 3x = 90 → x = 30 |
| 4 | m∠WOX = 130° | 90° + 40° = 130°, since OY ⊥ WZ and OX makes 40° with OY |

---

✘ Errors in Image:

- Problem 1: Incorrectly says B = 145°, but B is 35°
- Problem 4: Incorrectly says m∠WOX = 50°; should be 130°

---

✔ Final Answers:

1. ∠A = 55°
2. x = 39°
3. x = 30°
4. m∠WOX = 130°

Let me know if you'd like a visual explanation or corrected version of the image!