Geometry and proof · Ages 12–13

Isosceles triangles 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.Two sides of the triangle are equal, as marked. x =
Answer:
2.Two sides of the triangle are equal, as marked. x =
Answer:
3.Two sides of the triangle are equal and one side is extended, as shown. x =
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4.In the picture AB = AC, ∠BAC = 68°, and BC is extended to D. ∠ACB = °, ∠ACD = °.
Fill in each blank
Answer:
5.In the picture AB = AC, ∠BAC = 90°, and BC is extended to D. ∠ACB = °, ∠ACD = °.
Fill in each blank
Answer:
6.Two sides of the triangle are equal, and the line through its top is parallel to the base, as shown. x =
Answer:
7.In the picture CA = CB and ∠ACB = 42°. ∠BAC =
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8.In the picture CA = CB and ∠ACB = 110°. ∠BAC =
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9.In the picture AB = AC and ∠BAC = 104°. What is x?
- A. 38°
- B. 76°
- C. 142°
- D. 104°
Choose the right answer
Answer:
10.In the picture AB = AC and ∠BAC = 108°. What is x?
- A. 72°
- B. 144°
- C. 108°
- D. 36°
Choose the right answer
Answer:
11.An isosceles triangle has a top angle of 50° and base angles of 130°.
True or false?
Answer:
12.An isosceles triangle has a top angle of 70° and base angles of 55°.
True or false?
Answer:

13.AB = AC and ∠BAC = 36°. BC is extended to D. What is ∠ACD?
- A. 72°
- B. 108°
- C. 36°
- D. 144°
Choose the right answer
Working:
Answer:
14.A roof's cross-section is an isosceles triangle, and the beam along its top is parallel to the base, as shown. What is x?
Working:
Answer:
15.AB = AC and ∠BAC = 56°, as shown. ∠ABC = ∠ACB = 62° (______). Which reason goes in the blank?
- A. base ∠s, isos. △
- B. ext. ∠ of △
- C. vert. opp. ∠s
- D. ∠ sum of △
Choose the right answer
Working:
Answer:
16.AB = AC, BC is extended to D and ∠ACD = 134°, as shown. What is ∠BAC?
Working:
Answer:
17.An isosceles triangle's top angle is 27° more than a base angle. Base angle = °, top angle = °.
Fill in each blank
Working:
Answer:
18.AB = AC and ∠BAC = 128°, as shown. Find ∠ABC: put the steps in order.
- A. ∠ABC = ∠ACB (base ∠s, isos. △)
- B. 2∠ABC + 128° = 180° (∠ sum of △)
- C. ∠ABC = 26°
- D. 2∠ABC = 52°
Write the letters in the right order
Working:
Answer:

Show the answers and working

Isosceles triangles worksheet: Answers and working

1. 55°
The two base angles are equal (base ∠s, isos. △): x = (180° − 70°) ÷ 2 = 55° (∠ sum of △).
2. 70°
The two base angles are equal, both 55° (base ∠s, isos. △): x = 180° − 2 × 55° = 70° (∠ sum of △).
3. 38°
The base angle beside the exterior one = 180° − 109° = 71° (adj. ∠s on st. line); the base angles are equal (base ∠s, isos. △), so x = 180° − 2 × 71° = 38° (∠ sum of △).
4. 56, 124
∠ACB = (180° − 68°) ÷ 2 = 56° (base ∠s, isos. △), ∠ACD = 180° − 56° = 124° (adj. ∠s on st. line).
5. 45, 135
∠ACB = (180° − 90°) ÷ 2 = 45° (base ∠s, isos. △), ∠ACD = 180° − 45° = 135° (adj. ∠s on st. line).
6. 51°
Each base angle = (180° − 78°) ÷ 2 = 51° (base ∠s, isos. △); x and the left base angle are alternate, so x = 51° (alt. ∠s).
7. 69°
CA = CB, so the angles facing them, ∠ABC and ∠BAC, are equal (base ∠s, isos. △): ∠BAC = (180° − 42°) ÷ 2 = 69°. The equal angles are at A and B, not at B and C.
8. 35°
CA = CB, so the angles facing them, ∠ABC and ∠BAC, are equal (base ∠s, isos. △): ∠BAC = (180° − 110°) ÷ 2 = 35°. The equal angles are at A and B, not at B and C.

9. A. 38°
AB = AC, so the base angles are equal (base ∠s, isos. △): x = (180° − 104°) ÷ 2 = 38°.
10. D. 36°
AB = AC, so the base angles are equal (base ∠s, isos. △): x = (180° − 108°) ÷ 2 = 36°.
11. False
The two base angles are equal and make 180° with the top (∠ sum of △, base ∠s, isos. △): each base angle = (180° − 50°) ÷ 2 = 65°.
12. True
The two base angles are equal and make 180° with the top (∠ sum of △, base ∠s, isos. △): each base angle = (180° − 70°) ÷ 2 = 55°.

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

13. B. 108°
∠ACB = (180° − 36°) ÷ 2 = 72° (base ∠s, isos. △), ∠ACD = 180° − 72° = 108° (adj. ∠s on st. line). Or by the exterior angle: 36° + 72° = 108°.
14. 68°
56° and the right base angle are alternate, so that base angle = 56°; the base angles are equal (base ∠s, isos. △), so x = 180° − 2 × 56° = 68°.
15. A. base ∠s, isos. △
AB = AC, so the angles facing them are equal: the reason is base ∠s, isos. △. Then the angle sum: (180° − 56°) ÷ 2 = 62°.
16. 88°
∠ACB = 180° − 134° = 46° (adj. ∠s on st. line), ∠ABC = ∠ACB = 46° (base ∠s, isos. △), ∠BAC = 180° − 2 × 46° = 88° (∠ sum of △).
17. 51, 78
Let a base angle be y°: y + y + (y + 27) = 180 (∠ sum of △), 3y = 180 − 27, y = 51; top angle = 51° + 27° = 78°.
18. A → B → D → C
The base angles are equal, so the angle sum reads 2∠ABC + 128° = 180°, 2∠ABC = 52°, ∠ABC = 26°.

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