Geometry and proof · Ages 13–14

Proving a triangle isosceles worksheet

Part of: Congruent triangles

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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.In the picture, ∠B = ∠C. AC =
Answer:
2.In △ABC, ∠A = 76° and ∠B = 52°. Prove AB = AC: put the steps in order.
- A. ∠B = ∠C = 52°
- B. ∠C = 180° − 76° − 52° = 52° (∠ sum of △)
- C. AB = AC (sides opp. equal ∠s)
Write the letters in the right order
Answer:
3.In △ABC, ∠A = 84°, ∠C = 48°, AB = 9 cm and BC = 10 cm. AC =
Answer:
4.In △ABC, ∠A = 92°, ∠C = 44°, AB = 15 cm and BC = 8 cm. AC =
Answer:
5.In △ABC, ∠A = 50° and ∠B = 65°. Prove AB = AC: put the steps in order.
- A. ∠C = 180° − 50° − 65° = 65° (∠ sum of △)
- B. AB = AC (sides opp. equal ∠s)
- C. ∠B = ∠C = 65°
Write the letters in the right order
Answer:
6.In the picture BC is extended to D, ∠ACD = 114°, ∠BAC = 57° and AC = 5 cm. ∠ABC = °, BC = cm.
Fill in each blank
Answer:
7.In △ABC, BD bisects ∠ABC with D on AC; F is on AB with DF // BC, and BF = 8 cm. DF =
Answer:
8.In △ABC, BD bisects ∠ABC with D on AC; F is on AB with DF // BC, and BF = 11 cm. DF =
Answer:

13.In △ABC, ∠A = 31°, and BC is extended to D with ∠ACD = 149°. So AB = BC.
True or false?
Working:
Answer:
14.A flagpole AB stands upright at B on flat ground. From C on the ground, ∠ACB = 45° up to the top A, and BC = 35 m. How tall is the flagpole?
Working:
Answer:
15.In △ABC, BD bisects ∠ABC with D on AC; F is on AB with DF // BC, and BF = 3 cm. Find FD: put the steps in order.
- A. FD = FB (sides opp. equal ∠s)
- B. FD = 3 cm
- C. ∠FDB = ∠FBD (BD bisects ∠ABC)
- D. ∠FDB = ∠DBC (alt. ∠s, DF // BC)
Write the letters in the right order
Working:
Answer:
16.In △ABC, ∠A = 61° and ∠B = 58°. Which two sides must be equal?
- A. AC = BC
- B. No two sides are equal
- C. AB = BC
- D. AB = AC
Choose the right answer
Working:
Answer:
17.In △ABC, AC = BC, ∠A = (x + 20)° and ∠B = (2x − 30)°. x = , ∠C = °.
Fill in each blank
Working:
Answer:
18.In △ABC, AC = BC, ∠A = (x + 25)° and ∠B = (2x − 11)°. x = , ∠C = °.
Fill in each blank
Working:
Answer:

Show the answers and working

Proving a triangle isosceles worksheet: Answers and working

1. 5 cm
∠B faces AC and ∠C faces AB; ∠B = ∠C, so AC = AB = 5 cm (sides opp. equal ∠s).
2. B → A → C
∠C first; then ∠B = ∠C; last, sides opposite equal angles give AB = AC.
3. 9 cm
∠B = 180° − 84° − 48° = 48° (∠ sum of △), so ∠B = ∠C and AC = AB = 9 cm (sides opp. equal ∠s).
4. 15 cm
∠B = 180° − 92° − 44° = 44° (∠ sum of △), so ∠B = ∠C and AC = AB = 15 cm (sides opp. equal ∠s).
5. A → C → B
∠C first; then ∠B = ∠C; last, sides opposite equal angles give AB = AC.
6. 57, 5
∠ABC = 114° − 57° = 57° (ext. ∠ of △), so ∠ABC = ∠BAC and BC = AC = 5 cm (sides opp. equal ∠s).
7. 8 cm
∠FBD = ∠DBC (BD bisects ∠ABC) and ∠FDB = ∠DBC (alt. ∠s, DF // BC), so ∠FBD = ∠FDB and DF = BF = 8 cm (sides opp. equal ∠s).
8. 11 cm
∠FBD = ∠DBC (BD bisects ∠ABC) and ∠FDB = ∠DBC (alt. ∠s, DF // BC), so ∠FBD = ∠FDB and DF = BF = 11 cm (sides opp. equal ∠s).

9. A. AB = AC
Equal angles face equal sides: AC, facing ∠B, equals AB, facing ∠C.
10. C. AB = AC
Equal angles face equal sides: AC, facing ∠B, equals AB, facing ∠C.
11. False
∠C = 180° − 74° − 69° = 37°. Only with two equal angles are two sides equal (sides opp. equal ∠s), making it isosceles.
12. True
∠C = 180° − 50° − 65° = 65°. Only with two equal angles are two sides equal (sides opp. equal ∠s), making it isosceles.

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

13. True
∠ACB = 180° − 149° (adj. ∠s on st. line). AB and BC face ∠ACB and ∠A, and are equal only when those angles are.
14. 35 m
∠ABC = 90°, so ∠BAC = 180° − 90° − 45° = 45° = ∠ACB, and AB = BC = 35 m (sides opp. equal ∠s).
15. D → C → A → B
Alternate angles give ∠FDB = ∠DBC, and ∠DBC = ∠FBD, so ∠FDB = ∠FBD and FD = FB = 3 cm.
16. C. AB = BC
∠C = 180° − 61° − 58°. The two equal angles face the two equal sides (sides opp. equal ∠s).
17. 50, 40
AC = BC, so ∠A = ∠B (base ∠s, isos. △): x + 20 = 2x − 30, x = 50; ∠A = ∠B = 70°, ∠C = 180° − 2 × 70° = 40°.
18. 36, 58
AC = BC, so ∠A = ∠B (base ∠s, isos. △): x + 25 = 2x − 11, x = 36; ∠A = ∠B = 61°, ∠C = 180° − 2 × 61° = 58°.

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