Geometry and proof · Ages 12–13

Angles of a triangle worksheet

Part of: Angles & parallel lines

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Geometry and proof · Ages 12–13

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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.One side of the triangle is extended. x =
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2.BC is extended to D, as shown. ∠ACB = ° (∠ sum of △), ∠ACD = ° (adj. ∠s on st. line).
Fill in each blank
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3.Two angles of the triangle are marked. x =
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4.The picture shows a right-angled triangle. x =
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5.One side of the triangle is extended, and the exterior angle is marked. x =
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6.In △ABC, ∠A = x°, ∠B = 2x° and ∠C = (x + 48)°. ∠C =
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7.In △ABC, BC is extended to D. ∠BAC = (x + 31)°, ∠ABC = 4x° and the exterior angle ∠ACD = (3x + 67)°. ∠ACD =
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8.In △ABC, BC is extended to D. ∠BAC = (x + 33)°, ∠ABC = 4x° and the exterior angle ∠ACD = (3x + 79)°. ∠ACD =
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9.In △ABC, ∠A = x°, ∠B = 3x° and ∠C = (x + 30)°. ∠C =
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13.A triangle's three exterior angles (one at each corner) are in the ratio 3 : 7 : 5. What is its largest interior angle?
Working:
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14.In △ABC, ∠A = x°, ∠B = (x + 34)°, ∠C = (2x − 10)°. x + (x + 34) + (2x − 10) = 180, x = , the largest angle = °.
Fill in each blank
Working:
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15.One side of the triangle is extended, as shown. What is x?
- A. 81°
- B. 181°
- C. 49°
- D. 130°
Choose the right answer
Working:
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16.A triangular pane of glass has two angles of 86° and 36°. What is the third angle?
Working:
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17.In △ABC, ∠A : ∠B : ∠C = 2 : 2 : 1. ∠A = °, ∠B = °, ∠C = °.
Fill in each blank
Working:
Answer:
18.BC is extended to D, as shown. ∠ACD = ∠BAC + ∠ABC = 58° + 51° = 109° (______). Which reason goes in the blank?
- A. adj. ∠s on st. line
- B. alt. ∠s, AB // CD
- C. ∠ sum of △
- D. ext. ∠ of △
Choose the right answer
Working:
Answer:

Show the answers and working

Angles of a triangle worksheet: Answers and working

1. 147°
x is an exterior angle, equal to the two interior angles not next to it: x = 64° + 83° = 147° (ext. ∠ of △).
2. 37, 143
∠ACB = 180° − 89° − 54° = 37°, ∠ACD = 180° − 37° = 143°. Or straight from the exterior angle: 89° + 54° = 143° (ext. ∠ of △).
3. 43°
The three angles add to 180°: x = 180° − 47° − 90° = 43° (∠ sum of △).
4. 35°
x + 55° + 90° = 180° (∠ sum of △), x = 90° − 55° = 35°.
5. 79°
The exterior angle 137° equals the two far interior angles added: x + 58° = 137° (ext. ∠ of △), so x = 137° − 58° = 79°.
6. 81°
x + 2x + (x + 48) = 180 (∠ sum of △), 4x = 132, x = 33; ∠C = 33° + 48° = 81°.
7. 121°
The exterior angle equals the two far interior angles added: (x + 31) + 4x = 3x + 67 (ext. ∠ of △), 2x = 36, x = 18. ∠ACD = 3 × 18 + 67 = 121°.
8. 148°
The exterior angle equals the two far interior angles added: (x + 33) + 4x = 3x + 79 (ext. ∠ of △), 2x = 46, x = 23. ∠ACD = 3 × 23 + 79 = 148°.
9. 60°
x + 3x + (x + 30) = 180 (∠ sum of △), 5x = 150, x = 30; ∠C = 30° + 30° = 60°.

10. B. ∠ sum of △
An interior x: the three angles add to 180° (∠ sum of △); an exterior x: it equals the two far interior angles added (ext. ∠ of △). Here x = 91°.
11. False
A triangle's three angles add to 180° (∠ sum of △): 58° + 33° + 94° = 185°.
12. True
A triangle's three angles add to 180° (∠ sum of △): 80° + 92° + 8° = 180°.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 108°
The three exterior angles add to 360° (each is 180° − its interior angle: 3 × 180° − 180° = 360°), so each part is 24°. The largest interior angle sits by the smallest exterior one: 180° − 3 × 24° = 108°.
14. 39, 73
4x + 34 − 10 = 180 (∠ sum of △), 4x = 180 − 34 + 10, x = 39. The angles are 39°, x + 34 and 2x − 10; the largest is 73°.
15. A. 81°
The exterior angle 131° is x and 50° added (ext. ∠ of △): x = 131° − 50° = 81°.
16. 58°
A triangle's three angles add to 180° (∠ sum of △): 180° − 86° − 36° = 58°.
17. 72, 72, 36
The three add to 180° (∠ sum of △): 2 + 2 + 1 parts of 36° each, so 72°, 72°, 36°.
18. D. ext. ∠ of △
∠ACD is an exterior angle, equal to the two interior angles not next to it, ∠BAC and ∠ABC: the reason is ext. ∠ of △.

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