Geometry and proof · Ages 13–14
Properties of isosceles triangles worksheet
Part of: Congruent triangles
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Properties of isosceles triangles worksheet
Geometry and proof · Ages 13–14
Numsprite
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Name:Date:Score:／48
Write each answer in the space. For word problems, show your working and write a sentence answer.
Part A: Working out and filling in (2 marks each)
1.In the picture, AB = AC. x =
Answer:
2.In the picture, AB = AC, AD ⊥ BC and BC = 18 cm. BD =
Answer:
3.In the picture, AB = AC, AD bisects ∠BAC, and ∠BAC = 76°. ∠ADB =
Answer:
4.In the picture, AB = AC, AD bisects ∠BAC, and ∠BAC = 60°. ∠ADB =
Answer:
5.△ABC is equilateral with sides of 16 cm, and AD ⊥ BC. BD = cm, ∠BAD = °.
Fill in each blank
Answer:
6.In the picture, AB = AC, D is the midpoint of BC and ∠BAC = 86°. ∠BAD = °, ∠ADB = °, ∠ABD = °.
Fill in each blank
Answer:
7.△ABC is equilateral with sides of 14 cm, and AD ⊥ BC. BD = cm, ∠BAD = °.
Fill in each blank
Answer:
8.In the picture, AB = AC, D is the midpoint of BC and ∠BAC = 62°. ∠BAD = °, ∠ADB = °, ∠ABD = °.
Fill in each blank
Answer:
9.In the isosceles triangle ABC, AB = AC = 8 cm, AD ⊥ BC with D on BC, and BD = 3 cm. The perimeter of △ABC =
Answer:
10.In the picture, AB = AC and AD bisects ∠BAC. x =
Answer:
Part B: Choose, or true or false (2 marks each)
11.An isosceles triangle's top angle is 114°, and its base angles are 114°.
True or false?
Answer:
12.An isosceles triangle's top angle is 84°, and its base angles are 96°.
True or false?
Answer:
Part C: Word problems (4 marks each)
13.One angle of an isosceles triangle is 132°. What are the other two?
- A. both 132°
- B. both 90°
- C. both 24°
- D. both 48°
Choose the right answer
Working:
Answer:
14.In △ABC, AB = AC and AD is the median to the base BC. ∠BAC = 40°, so ∠BAD = 40°.
True or false?
Working:
Answer:
15.An isosceles triangle's base angles are each 24° more than its top angle. What is the top angle?
Working:
Answer:
16.In a roof frame ABC, as shown, AB = AC and the upright AD ⊥ BC. BC = 28 cm and ∠BAC = 94°. BD = cm, ∠BAD = °.
Fill in each blank
Working:
Answer:
17.An isosceles triangle has a perimeter of 70 cm and a base of 32 cm. How long is each of its equal sides?
Working:
Answer:
18.In △ABC, AB = AC and ∠BAC = 54°. Find ∠ABC: put the steps in order.
- A. ∠ABC = 63°
- B. 2∠ABC = 126°
- C. 2∠ABC + 54° = 180° (∠ sum of △)
- D. ∠ABC = ∠ACB (base ∠s, isos. △)
Write the letters in the right order
Working:
Answer:
Show the answers and working
Properties of isosceles triangles worksheet: Answers and working
Part A: Working out and filling in
1. 61°
∠B = ∠C (base ∠s, isos. △): x = (180° − 58°) ÷ 2 = 61° (∠ sum of △).
2. 9 cm
In an isosceles triangle the height to the base is also its median (prop. of isos. △): BD = 18 ÷ 2 = 9 cm.
3. 90°
In an isosceles triangle the bisector of the top angle is also the height to the base (prop. of isos. △): AD ⊥ BC, ∠ADB = 90°, whatever ∠BAC (76°) is.
4. 90°
In an isosceles triangle the bisector of the top angle is also the height to the base (prop. of isos. △): AD ⊥ BC, ∠ADB = 90°, whatever ∠BAC (60°) is.
5. 8, 30
An equilateral triangle is isosceles too: the height AD is also the median and the bisector, so BD = 16 ÷ 2 = 8 cm and ∠BAD = 60° ÷ 2 = 30°.
6. 43, 90, 47
The median to the base is also the bisector of the top angle and the height (prop. of isos. △): ∠BAD = 86° ÷ 2 = 43°, ∠ADB = 90°, ∠ABD = 180° − 90° − 43° = 47°.
7. 7, 30
An equilateral triangle is isosceles too: the height AD is also the median and the bisector, so BD = 14 ÷ 2 = 7 cm and ∠BAD = 60° ÷ 2 = 30°.
8. 31, 90, 59
The median to the base is also the bisector of the top angle and the height (prop. of isos. △): ∠BAD = 62° ÷ 2 = 31°, ∠ADB = 90°, ∠ABD = 180° − 90° − 31° = 59°.
9. 22 cm
The height to the base is also its median (prop. of isos. △): DC = BD = 3 cm, BC = 6 cm; perimeter = 8 + 8 + 6 = 22 cm.
10. 43°
The bisector of the top angle is at right angles to BC (prop. of isos. △), ∠ADB = 90°: x = 180° − 90° − 47° = 43° (∠ sum of △).
Part B: Choose, or true or false
11. False
The two base angles are equal and make 180° with the top: each = (180° − 114°) ÷ 2 = 33°.
12. False
The two base angles are equal and make 180° with the top: each = (180° − 84°) ÷ 2 = 48°.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. C. both 24°
132° is obtuse, so it must be the top angle; each base angle = (180° − 132°) ÷ 2 = 24°.
14. False
The median to the base also bisects the top angle (prop. of isos. △): ∠BAD = 40° ÷ 2 = 20°.
15. 44°
Let the top angle be x: x + 2(x + 24°) = 180°, 3x = 180° − 2 × 24°, x = 44° (base angles 68°).
16. 14, 47
The height to the base is also the median and the bisector (prop. of isos. △): BD = 28 ÷ 2 = 14 cm, ∠BAD = 94° ÷ 2 = 47°.
17. 19 cm
The two equal sides share the rest: (70 − 32) ÷ 2 = 19 cm.
18. D → C → B → A
The base angles are equal and the three make 180°: 2∠ABC = 180° − 54° = 126°, ∠ABC = 63°.
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