Math worksheet for solving algebraic equations using angle relationships on a straight line.

You're given 8 problems where you need to find the values of variables a and b, using the fact that angle QRL is 180° — meaning the angles on a straight line at point Q add up to 180°.

In each diagram, two angles are adjacent and form a straight line (linear pair), so their measures must sum to 180°.

We’ll solve each one step by step.

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Problem 1:

Angles:
- ∠XQM = -3a + 56
- ∠MQR = 3b + 85
They form a straight line → sum = 180°

So:
(-3a + 56) + (3b + 85) = 180

Simplify:
-3a + 3b + 141 = 180
→ -3a + 3b = 39
Divide by 3:
→ -a + b = 13 → Equation (1)

Also given: angle XQL = 157 - 3b
But since XQ and QR are opposite rays, and M and L are on opposite sides, we can also consider angle XQL and angle LQR as linear pair? Wait — actually, looking at the diagram, it seems angle XQL and angle MQR might be vertical or something else.

Wait — let’s re-express.

Actually, in Problem 1, we have:

- Angle XQM = -3a + 56
- Angle MQR = 3b + 85
Sum = 180° → already used.

Also, angle XQL = 157 - 3b
And angle LQR = 5a + 40

Since XQ and QR are straight, then angle XQL + angle LQR should also equal 180°.

So:

(157 - 3b) + (5a + 40) = 180

→ 5a - 3b + 197 = 180
→ 5a - 3b = -17 → Equation (2)

Now solve system:

From (1): b = a + 13

Plug into (2):

5a - 3(a + 13) = -17
5a - 3a - 39 = -17
2a = 22
→ a = 11

Then b = 11 + 13 = 24

✔ Answer 1: a=11, b=24

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Problem 2:

Angles:

- ∠XQM = -10a + 237
- ∠MQR = 5b + 63
Sum = 180°

So:
(-10a + 237) + (5b + 63) = 180
→ -10a + 5b + 300 = 180
→ -10a + 5b = -120
Divide by 5:
→ -2a + b = -24 → Equation (1)

Also given:

- ∠XQL = 99 - 14b
- ∠LQR = -4a + 177
Sum = 180°

(99 - 14b) + (-4a + 177) = 180
→ -4a -14b + 276 = 180
→ -4a -14b = -96
Divide by 2:
→ -2a -7b = -48 → Equation (2)

Now subtract Equation (1) from Equation (2):

Eq(2): -2a -7b = -48
Eq(1): -2a + b = -24
Subtract:
(-2a -7b) - (-2a + b) = -48 - (-24)
→ -8b = -24
→ b = 3

Plug into Eq(1): -2a + 3 = -24 → -2a = -27 → a = 13.5

✔ Answer 2: a=13.5, b=3

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Problem 3:

Angles:

- ∠XQM = 4b + 147
- ∠MQR = 5b + 11
Sum = 180°

So:
4b + 147 + 5b + 11 = 180
→ 9b + 158 = 180
→ 9b = 22
→ b = 22/9 ≈ 2.444...

Wait — but we also have:

∠XQL = 35 + 2b
∠LQR = -2a + 135
Sum = 180°

So:
35 + 2b -2a + 135 = 180
→ -2a + 2b + 170 = 180
→ -2a + 2b = 10
→ -a + b = 5 → Equation (1)

We already have b = 22/9

Then: -a + 22/9 = 5
→ -a = 5 - 22/9 = (45 - 22)/9 = 23/9
→ a = -23/9

✔ Answer 3: a = -23/9, b = 22/9

*(Note: Fractions are acceptable unless specified otherwise)*

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Problem 4:

Angles:

- ∠XQM = 1a + 141
- ∠MQR = -1b + 5
Sum = 180°

So:
a + 141 - b + 5 = 180
→ a - b + 146 = 180
→ a - b = 34 → Equation (1)

Also:

∠XQL = 107 - 11b
∠LQR = 5a + 105
Sum = 180°

So:
107 - 11b + 5a + 105 = 180
→ 5a -11b + 212 = 180
→ 5a -11b = -32 → Equation (2)

From (1): a = b + 34

Plug into (2):

5(b + 34) -11b = -32
5b + 170 -11b = -32
-6b = -202
→ b = 202/6 = 101/3 ≈ 33.666...

Then a = 101/3 + 34 = 101/3 + 102/3 = 203/3

✔ Answer 4: a = 203/3, b = 101/3

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Problem 5:

Angles:

- ∠XQM = -1a + 130
- ∠MQR = 3b + 43
Sum = 180°

So:
-a + 130 + 3b + 43 = 180
→ -a + 3b + 173 = 180
→ -a + 3b = 7 → Equation (1)

Also:

∠XQL = -6b + 79
∠LQR = 4a + 105
Sum = 180°

So:
-6b + 79 + 4a + 105 = 180
→ 4a -6b + 184 = 180
→ 4a -6b = -4
Divide by 2:
→ 2a - 3b = -2 → Equation (2)

Now use Eq(1): -a + 3b = 7 → multiply by 2: -2a + 6b = 14

Add to Eq(2):
(2a - 3b) + (-2a + 6b) = -2 + 14
→ 3b = 12 → b = 4

Plug into Eq(1): -a + 3(4) = 7 → -a + 12 = 7 → a = 5

✔ Answer 5: a=5, b=4

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Problem 6:

Angles:

- ∠XQM = -5a + 149
- ∠MQR = 16 + 5b
Sum = 180°

So:
-5a + 149 + 16 + 5b = 180
→ -5a + 5b + 165 = 180
→ -5a + 5b = 15
Divide by 5:
→ -a + b = 3 → Equation (1)

Also:

∠XQL = -1b + 46
∠LQR = 3a + 139
Sum = 180°

So:
-b + 46 + 3a + 139 = 180
→ 3a - b + 185 = 180
→ 3a - b = -5 → Equation (2)

Now from (1): b = a + 3

Plug into (2):

3a - (a + 3) = -5
3a - a - 3 = -5
2a = -2
→ a = -1

Then b = -1 + 3 = 2

✔ Answer 6: a=-1, b=2

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Problem 7:

Angles:

- ∠XQM = -4a + 123
- ∠MQR = 101 - 4b
Sum = 180°

So:
-4a + 123 + 101 - 4b = 180
→ -4a -4b + 224 = 180
→ -4a -4b = -44
Divide by -4:
→ a + b = 11 → Equation (1)

Also:

∠XQL = 109 - 6b
∠LQR = 2a + 81
Sum = 180°

So:
109 - 6b + 2a + 81 = 180
→ 2a -6b + 190 = 180
→ 2a -6b = -10
Divide by 2:
→ a - 3b = -5 → Equation (2)

From (1): a = 11 - b

Plug into (2):

11 - b - 3b = -5
11 - 4b = -5
-4b = -16
→ b = 4

Then a = 11 - 4 = 7

✔ Answer 7: a=7, b=4

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Problem 8:

Angles:

- ∠XQM = -9a + 139
- ∠MQR = 10b + 5b = 15b? Wait — looks like it says “10b + 5b” — probably typo? Let me check image description.

Actually, in your text:
“-9a + 139” and “10b + 5b” — that would be 15b. But maybe it’s “10b + 5”?

Looking back at your original text:

> 8)
> -9a + 139
> 10b + 5b
> 3b + 116
> -1a + 59

That seems inconsistent. Probably it's:

- ∠XQM = -9a + 139
- ∠MQR = 10b + 5 → (assuming “+5b” was meant to be “+5”) — because 10b + 5b = 15b doesn’t make sense with other terms.

Alternatively, maybe it’s “10b + 5” — let’s assume that for now.

Also, ∠XQL = 3b + 116
∠LQR = -1a + 59
Sum = 180°

First, use ∠XQM + ∠MQR = 180°

Assume: ∠MQR = 10b + 5

Then:
(-9a + 139) + (10b + 5) = 180
→ -9a + 10b + 144 = 180
→ -9a + 10b = 36 → Equation (1)

Now, ∠XQL + ∠LQR = 180°

(3b + 116) + (-a + 59) = 180
→ -a + 3b + 175 = 180
→ -a + 3b = 5 → Equation (2)

From (2): a = 3b - 5

Plug into (1):

-9(3b - 5) + 10b = 36
→ -27b + 45 + 10b = 36
→ -17b = -9
→ b = 9/17

Then a = 3*(9/17) - 5 = 27/17 - 85/17 = -58/17

✔ Answer 8: a = -58/17, b = 9/17

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## ✔ Final Answers:

1. a = 11, b = 24
2. a = 13.5, b = 3
3. a = -23/9, b = 22/9
4. a = 203/3, b = 101/3
5. a = 5, b = 4
6. a = -1, b = 2
7. a = 7, b = 4
8. a = -58/17, b = 9/17

*(Note: For problem 8, if "10b + 5b" was intended to mean 15b, then:*

*(-9a + 139) + 15b = 180 → -9a + 15b = 41*
*and from other pair: -a + 3b = 5 → a = 3b - 5*
*Then: -9(3b - 5) + 15b = 41 → -27b + 45 + 15b = 41 → -12b = -4 → b = 1/3, a = 3*(1/3) - 5 = 1 - 5 = -4*

*So if it's 15b, answer is a=-4, b=1/3 — but based on context, I think “10b + 5” is more likely. Since the worksheet shows “10b + 5b”, it may be a typo. But without seeing the image, I’ll stick with my first interpretation.)*

If you can confirm the expression for problem 8, I can adjust!

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📌 Summary Table:

| Problem | a | b |
|---------|------------|------------|
| 1 | 11 | 24 |
| 2 | 13.5 | 3 |
| 3 | -23/9 | 22/9 |
| 4 | 203/3 | 101/3 |
| 5 | 5 | 4 |
| 6 | -1 | 2 |
| 7 | 7 | 4 |
| 8 | -58/17 | 9/17 |

Let me know if you’d like decimal approximations or if you want to verify problem 8 with the actual image!