Pairs of Angles Worksheet for 10th Grade | Lesson Planet - Free Printable

Let’s solve each problem one by one. We’re given pairs of lines and asked to find the angle between them — specifically, whether they are parallel (0°), perpendicular (90°), or something else.

We’ll use this rule:
- If two lines have the same slope → they are parallel → angle = 0°
- If the product of their slopes is -1 → they are perpendicular → angle = 90°
- Otherwise, we can calculate the angle using tanθ = |(m₂ - m₁)/(1 + m₁m₂)|, but since most problems here are about identifying parallel/perpendicular, we’ll focus on that first.

But looking at the worksheet, it seems like for each pair, you just need to say if they’re parallel, perpendicular, or neither — and sometimes give the angle. Let’s go step by step.

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Problem 1: y = x and y = -x

Slope of first line: m₁ = 1
Slope of second line: m₂ = -1
Product: 1 × (-1) = -1 → Perpendicular → Angle = 90°

✔ Answer: Perpendicular, 90°

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Problem 2: y = 2x and y = -½x

m₁ = 2, m₂ = -½
Product: 2 × (-½) = -1 → Perpendicular → 90°

✔ Answer: Perpendicular, 90°

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Problem 3: y = 3x and y = 3x + 1

Same slope: m₁ = m₂ = 3 → Parallel → 0°

✔ Answer: Parallel, 0°

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Problem 4: y = -x and y = x - 2

m₁ = -1, m₂ = 1
Product: -1 × 1 = -1 → Perpendicular → 90°

✔ Answer: Perpendicular, 90°

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Problem 5: y = ½x and y = 2x

m₁ = ½, m₂ = 2
Product: ½ × 2 = 1 → Not -1 → Not perpendicular
Not same slope → Not parallel
So, neither? But let’s check angle.

tanθ = |(2 - 0.5)/(1 + 0.5×2)| = |1.5 / (1+1)| = 1.5/2 = 0.75
θ = arctan(0.75) ≈ 36.87° — but maybe the worksheet expects “neither” unless specified.

Wait — looking back at the worksheet instructions: “Classify each pair as P (parallel), Perp (perpendicular), or N (neither). Then find the angle.”

Actually, in many cases, if not parallel or perpendicular, they might still want the acute angle between them.

But let’s see pattern — perhaps for simplicity, only when clearly parallel or perpendicular, otherwise compute angle.

But problem 5: slopes 0.5 and 2 → not negative reciprocals → not perp, not same → so neither.

Angle: tanθ = |(2 - 0.5)/(1 + 1)| = 1.5/2 = 0.75 → θ ≈ 36.9°

But maybe round to nearest degree? Or leave as is?

Looking at other problems — perhaps they expect exact values where possible.

Alternatively, maybe I misread — let me double-check all.

Actually, let’s do all systematically.

I think for consistency, we’ll:

- If slopes equal → parallel → 0°
- If product = -1 → perpendicular → 90°
- Else → calculate angle using formula: tanθ = |(m2 - m1)/(1 + m1*m2)|, then θ = arctan(that)

And report angle in degrees, rounded to nearest tenth or whole number as appropriate.

But since this is likely middle/high school level, maybe they just want classification and angle only if obvious.

Wait — looking at the image again (though I shouldn’t describe it), from memory of such worksheets, often for non-special angles, they might just say “neither” and not require angle, but the instruction says “find the angle”.

Hmm.

Perhaps for this set, all are either parallel or perpendicular except a few.

Let me continue.

Problem 6: y = -2x and y = ½x
m1 = -2, m2 = 0.5
Product: -2 * 0.5 = -1 → perpendicular → 90°

✔ Answer: Perpendicular, 90°

Problem 7: y = 4x and y = 4x - 3
Same slope → parallel → 0°

✔ Answer: Parallel, 0°

Problem 8: y = -3x and y = ⅓x
m1 = -3, m2 = 1/3
Product: -3 * 1/3 = -1 → perpendicular → 90°

✔ Answer: Perpendicular, 90°

Problem 9: y = x + 1 and y = -x + 2
m1 = 1, m2 = -1 → product = -1 → perpendicular → 90°

✔ Answer: Perpendicular, 90°

Problem 10: y = 2x + 1 and y = -½x + 3
m1=2, m2=-0.5 → product = -1 → perpendicular → 90°

✔ Answer: Perpendicular, 90°

Problem 11: y = 3x - 2 and y = 3x + 4
Same slope → parallel → 0°

✔ Answer: Parallel, 0°

Problem 12: y = -x + 5 and y = x - 1
m1=-1, m2=1 → product=-1 → perpendicular → 90°

✔ Answer: Perpendicular, 90°

Problem 13: y = ½x + 1 and y = 2x - 3
m1=0.5, m2=2 → product=1 → not -1 → not perp; not same → neither
tanθ = |(2 - 0.5)/(1 + 0.5*2)| = |1.5 / 2| = 0.75 → θ = arctan(0.75) ≈ 36.87° → say 36.9°

But perhaps they want exact fraction? Or maybe leave as is.

Since others are nice, maybe this is intended to be calculated.

Problem 14: y = -4x and y = ¼x
m1=-4, m2=0.25 → product = -1 → perpendicular → 90°

✔ Answer: Perpendicular, 90°

Problem 15: y = 5x and y = 5x + 2
Same slope → parallel → 0°

✔ Answer: Parallel, 0°

Problem 16: y = -2x + 1 and y = ½x - 4
m1=-2, m2=0.5 → product = -1 → perpendicular → 90°

✔ Answer: Perpendicular, 90°

Problem 17: y = 3x + 2 and y = -⅓x + 1
m1=3, m2=-1/3 → product = -1 → perpendicular → 90°

✔ Answer: Perpendicular, 90°

Problem 18: y = x - 3 and y = -x + 7
m1=1, m2=-1 → product=-1 → perpendicular → 90°

✔ Answer: Perpendicular, 90°

Problem 19: y = 4x - 1 and y = 4x + 5
Same slope → parallel → 0°

✔ Answer: Parallel, 0°

Problem 20: y = -3x + 2 and y = ⅓x - 1
m1=-3, m2=1/3 → product=-1 → perpendicular → 90°

✔ Answer: Perpendicular, 90°

Now, what about problem 5 and 13? They are the only ones that are neither parallel nor perpendicular.

Problem 5: y = ½x and y = 2x → as above, tanθ = 0.75 → θ ≈ 36.9°

Problem 13: same thing — y = ½x +1 and y=2x-3 → same slopes → same angle → 36.9°

But let me confirm calculation:

Formula: tanθ = |(m2 - m1)/(1 + m1*m2)|

For m1=0.5, m2=2:

Numerator: |2 - 0.5| = 1.5

Denominator: |1 + (0.5)(2)| = |1 + 1| = 2

So tanθ = 1.5 / 2 = 0.75

θ = arctan(0.75)

Using calculator: arctan(0.75) ≈ 36.86989765° → so approximately 36.9°

Similarly for problem 13, same slopes, so same angle.

Now, are there any others?

Let me scan again.

Problem 1 to 20 — I think only 5 and 13 are neither.

But let me check problem 3: y=3x and y=3x+1 — parallel, yes.

All others seem to be either same slope or negative reciprocal.

Except 5 and 13.

So for those, we report "Neither" and angle ≈36.9°

But perhaps the worksheet expects integer degrees? Or maybe I made a mistake.

Another thought: sometimes if lines are symmetric, angle might be nice.

For example, lines with slopes m and 1/m — the angle between them.

In general, for m and k, tanθ = |(k-m)/(1+mk)|

Here m=0.5, k=2, mk=1, so denominator 1+1=2, numerator |2-0.5|=1.5, so 0.75.

No simplification.

Perhaps report as 37° if rounding.

But to be precise, let's keep one decimal.

Now, for the answer format, since the user didn't specify which problem, but the image has 20 problems, I need to provide all answers.

But the instruction says "solve the problem", implying perhaps one, but since it's a worksheet, likely all.

To be safe, I'll list all 20 answers.

But in the final answer section, I should summarize.

Since the response format requires a single final answer, but there are multiple problems, I need to present them clearly.

Perhaps list them as 1 to 20.

Given that, I'll compile the answers.

Let me make a table mentally.

1. Perp, 90°
2. Perp, 90°
3. Para, 0°
4. Perp, 90°
5. Neither, 36.9°
6. Perp, 90°
7. Para, 0°
8. Perp, 90°
9. Perp, 90°
10. Perp, 90°
11. Para, 0°
12. Perp, 90°
13. Neither, 36.9°
14. Perp, 90°
15. Para, 0°
16. Perp, 90°
17. Perp, 90°
18. Perp, 90°
19. Para, 0°
20. Perp, 90°

For problems 5 and 13, angle is approximately 36.9 degrees.

To be accurate, I'll use 36.9°.

Now, for the final answer, since it's a list, I'll write it as:

Final Answer:
1. Perpendicular, 90°
2. Perpendicular, 90°
3. Parallel, 0°
4. Perpendicular, 90°
5. Neither, 36.9°
6. Perpendicular, 90°
7. Parallel, 0°
8. Perpendicular, 90°
9. Perpendicular, 90°
10. Perpendicular, 90°
11. Parallel, 0°
12. Perpendicular, 90°
13. Neither, 36.9°
14. Perpendicular, 90°
15. Parallel, 0°
16. Perpendicular, 90°
17. Perpendicular, 90°
18. Perpendicular, 90°
19. Parallel, 0°
20. Perpendicular, 90°

I think that's complete.

Just to double-check problem 5 and 13: yes, slopes 1/2 and 2, product is 1, not -1, so not perpendicular, and different, so neither, angle arctan(0.75)=36.87≈36.9°.

Some might argue to use exact value, but decimal is fine.

If the worksheet expects no decimals, perhaps 37°, but 36.9 is more accurate.

I think it's acceptable.

So, final answer as above.