Interior Angles of Triangles worksheet with algebraic expressions for angles in triangles.

Absolutely! Let’s solve each problem step by step using the Triangle Angle Sum Theorem, which states:

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

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Problem 1: Triangle ABC

Angles:
- ∠A = 4x
- ∠B = 7x – 2
- ∠C = 5x – 10

Equation & Work:
Sum of angles = 180°
→ 4x + (7x – 2) + (5x – 10) = 180
Combine like terms:
→ 4x + 7x + 5x – 2 – 10 = 180
→ 16x – 12 = 180
Add 12 to both sides:
→ 16x = 192
Divide by 16:
→ x = 12

Angle Measures:
- ∠A = 4x = 4(12) = 48°
- ∠B = 7x – 2 = 7(12) – 2 = 84 – 2 = 82°
- ∠C = 5x – 10 = 5(12) – 10 = 60 – 10 = 50°

✔ Check: 48 + 82 + 50 = 180 ✔️

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Problem 2: Triangle DEF

Angles:
- ∠D = 3x
- ∠E = 18x
- ∠F = 5x – 2

Equation & Work:
→ 3x + 18x + (5x – 2) = 180
→ 26x – 2 = 180
→ 26x = 182
→ x = 7

Angle Measures:
- ∠D = 3x = 3(7) = 21°
- ∠E = 18x = 18(7) = 126°
- ∠F = 5x – 2 = 5(7) – 2 = 35 – 2 = 33°

✔ Check: 21 + 126 + 33 = 180 ✔️

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Problem 3: Triangle GHI

Angles:
- ∠G = 16x
- ∠H = 16x
- ∠I = 13x

Equation & Work:
→ 16x + 16x + 13x = 180
→ 45x = 180
→ x = 4

Angle Measures:
- ∠G = 16x = 16(4) = 64°
- ∠H = 16x = 64°
- ∠I = 13x = 13(4) = 52°

✔ Check: 64 + 64 + 52 = 180 ✔️

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Problem 4: Triangle JKL

Angles:
- ∠J = x + 8
- ∠K = 6x – 5
- ∠L = 2x – 3

Equation & Work:
→ (x + 8) + (6x – 5) + (2x – 3) = 180
→ x + 6x + 2x + 8 – 5 – 3 = 180
→ 9x + 0 = 180
→ 9x = 180
→ x = 20

Angle Measures:
- ∠J = x + 8 = 20 + 8 = 28°
- ∠K = 6x – 5 = 6(20) – 5 = 120 – 5 = 115°
- ∠L = 2x – 3 = 2(20) – 3 = 40 – 3 = 37°

✔ Check: 28 + 115 + 37 = 180 ✔️

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Problem 5: Triangle MNO

Given:
- ∠N = 5 × ∠M
- ∠O = 4 × ∠M

Let ∠M = x
Then:
- ∠N = 5x
- ∠O = 4x

Equation & Work:
→ x + 5x + 4x = 180
→ 10x = 180
→ x = 18

Angle Measures:
- ∠M = x = 18°
- ∠N = 5x = 5(18) = 90°
- ∠O = 4x = 4(18) = 72°

✔ Check: 18 + 90 + 72 = 180 ✔️

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## ✔ Final Answers Summary:

| Problem | ∠A/∠D/∠G/∠J/∠M | ∠B/∠E/∠H/∠K/∠N | ∠C/∠F/∠I/∠L/∠O |
|---------|------------------|------------------|------------------|
| 1 | 48° | 82° | 50° |
| 2 | 21° | 126° | 33° |
| 3 | 64° | 64° | 52° |
| 4 | 28° | 115° | 37° |
| 5 | 18° | 90° | 72° |

You can now fill in your worksheet with these values and show the work as demonstrated above. Let me know if you’d like a printable version or explanations for younger students!