Geometry and proof · Ages 13–14

Isosceles triangles worksheet

Part of: Polygons

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Geometry and proof · Ages 13–14

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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.In the picture AB = AC, ∠BAC = 78°, and BC is extended to D. ∠ACB = °, ∠ACD = °.
Fill in each blank
Answer:
4.Two sides of the triangle are equal and one side is extended, as shown. x =
Answer:
5.In the picture AB = AC, ∠BAC = 112°, 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 = 78°. ∠BAC =
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8.In the picture CA = CB and ∠ACB = 80°. ∠BAC =
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9.In the picture AB = AC and ∠BAC = 72°. What is x?
- A. 72°
- B. 126°
- C. 54°
- D. 108°
Choose the right answer
Answer:
10.In the picture AB = AC and ∠BAC = 98°. What is x?
- A. 98°
- B. 139°
- C. 41°
- D. 82°
Choose the right answer
Answer:
11.An isosceles triangle has a top angle of 62° and base angles of 69°.
True or false?
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12.An isosceles triangle has a top angle of 132° and base angles of 24°.
True or false?
Answer:

13.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:
14.AB = AC and ∠BAC = 120°. BC is extended to D. What is ∠ACD?
- A. 60°
- B. 30°
- C. 150°
- D. 120°
Choose the right answer
Working:
Answer:
15.An isosceles triangle's top angle is 45° more than a base angle. Base angle = °, top angle = °.
Fill in each blank
Working:
Answer:
16.AB = AC and ∠BAC = 128°, as shown. ∠ABC = ∠ACB = 26° (______). Which reason goes in the blank?
- A. ext. ∠ of △
- B. vert. opp. ∠s
- C. ∠ sum of △
- D. base ∠s, isos. △
Choose the right answer
Working:
Answer:
17.One angle of an isosceles triangle is 24°. The other two can be 24° and °, or ° and °.
Fill in each blank
Working:
Answer:
18.One angle of an isosceles triangle is 150°. How big is each of the other two?
Working:
Answer:

Show the answers and working

Isosceles triangles worksheet: Answers and working

1. 72°
The two base angles are equal (base ∠s, isos. △): x = (180° − 36°) ÷ 2 = 72° (∠ sum of △).
2. 66°
The two base angles are equal, both 57° (base ∠s, isos. △): x = 180° − 2 × 57° = 66° (∠ sum of △).
3. 51, 129
∠ACB = (180° − 78°) ÷ 2 = 51° (base ∠s, isos. △), ∠ACD = 180° − 51° = 129° (adj. ∠s on st. line).
4. 104°
The base angle beside the exterior one = 180° − 142° = 38° (adj. ∠s on st. line); the base angles are equal (base ∠s, isos. △), so x = 180° − 2 × 38° = 104° (∠ sum of △).
5. 34, 146
∠ACB = (180° − 112°) ÷ 2 = 34° (base ∠s, isos. △), ∠ACD = 180° − 34° = 146° (adj. ∠s on st. line).
6. 42°
Each base angle = (180° − 96°) ÷ 2 = 42° (base ∠s, isos. △); x and the left base angle are alternate, so x = 42° (alt. ∠s).
7. 51°
CA = CB, so the angles facing them, ∠ABC and ∠BAC, are equal (base ∠s, isos. △): ∠BAC = (180° − 78°) ÷ 2 = 51°. The equal angles are at A and B, not at B and C.
8. 50°
CA = CB, so the angles facing them, ∠ABC and ∠BAC, are equal (base ∠s, isos. △): ∠BAC = (180° − 80°) ÷ 2 = 50°. The equal angles are at A and B, not at B and C.

9. C. 54°
AB = AC, so the base angles are equal (base ∠s, isos. △): x = (180° − 72°) ÷ 2 = 54°.
10. C. 41°
AB = AC, so the base angles are equal (base ∠s, isos. △): x = (180° − 98°) ÷ 2 = 41°.
11. False
The two base angles are equal and make 180° with the top (∠ sum of △, base ∠s, isos. △): each base angle = (180° − 62°) ÷ 2 = 59°.
12. True
The two base angles are equal and make 180° with the top (∠ sum of △, base ∠s, isos. △): each base angle = (180° − 132°) ÷ 2 = 24°.

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

13. 34°
73° and the right base angle are alternate, so that base angle = 73°; the base angles are equal (base ∠s, isos. △), so x = 180° − 2 × 73° = 34°.
14. C. 150°
∠ACB = (180° − 120°) ÷ 2 = 30° (base ∠s, isos. △), ∠ACD = 180° − 30° = 150° (adj. ∠s on st. line). Or by the exterior angle: 120° + 30° = 150°.
15. 45, 90
Let a base angle be y°: y + y + (y + 45) = 180 (∠ sum of △), 3y = 180 − 45, y = 45; top angle = 45° + 45° = 90°.
16. D. base ∠s, isos. △
AB = AC, so the angles facing them are equal: the reason is base ∠s, isos. △. Then the angle sum: (180° − 128°) ÷ 2 = 26°.
17. 132, 78, 78
Two cases: if 24° is a base angle, so is the other, and the top = 180° − 2 × 24° = 132°; if 24° is the top, each base angle = (180° − 24°) ÷ 2 = 78°.
18. 15°
150° is obtuse, and no triangle has two obtuse angles, so it must be the top angle. The base angles are equal: (180° − 150°) ÷ 2 = 15°.

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