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Geometry and proof · Ages 13–14

Congruent triangles worksheet

Covers: Congruent figures and transformations, Corresponding parts of congruent triangles, Conditions for congruent triangles, Proving triangles congruent, Properties of isosceles triangles, Proving a triangle isosceles, Bisecting an angle and a line segment, Angles compasses can draw

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Congruent 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. 1.△ABC is reflected to give △A′B′C′, then △A′B′C′ is enlarged 3 times to give △A″B″C″. AB = 4 cm, the perimeter of △ABC is 12 cm, and ∠B = 100°. A″B″ = cm, the perimeter of △A″B″C″ = cm, and ∠B″ = °.

    Fill in each blank

    Answer:

  2. 2.The two triangles are congruent; sides with the same marks are equal. x = , y = .

    Fill in each blank

    Answer:

  3. 3.In the picture, AB = AD, CB = CD and ∠BAC = 38°. Find ∠BAD: put the steps in order.

    • A. ∠BAD = 38° + 38° = 76°
    • B. AB = AD, CB = CD, AC = AC, so △ABC ≅ △ADC (SSS)
    • C. ∠DAC = ∠BAC = 38° (corr. ∠s, ≅△s)

    Write the letters in the right order

    Answer:

  4. 4.In the picture, AB = AD, CB = CD, ∠BAD = 68° and ∠BCD = 80°. ∠BAC = °, ∠ABC = °.

    Fill in each blank

    Answer:

  5. 5.In the isosceles triangle ABC, AB = AC = 15 cm, AD ⊥ BC with D on BC, and BD = 11 cm. The perimeter of △ABC =

    Answer:

  6. 6.In the picture BC is extended to D, ∠ACD = 98°, ∠BAC = 49° and AC = 6 cm. ∠ABC = °, BC = cm.

    Fill in each blank

    Answer:

  7. 7.Draw the perpendicular bisector of the line segment AB (8 cm) with compasses and a straight edge: put the steps in order.

    • A. AM = MB = 4 cm: join PQ, cutting AB at M
    • B. AQ = BQ: with the same radius, draw arcs again below AB, crossing at Q
    • C. AP = BP: with centres A and B and one radius (more than 4 cm), draw arcs crossing above AB at P

    Write the letters in the right order

    Answer:

  8. 8.A 60° angle is drawn and bisected 2 times; next to it, a 60° angle is drawn and bisected 1 times. Together the two make °

    Answer:

Part B: Choose, or true or false (2 marks each)

  1. 9.A square of side 5 cm is transformed, and its image has sides of 6 cm. The transformation could be a rotation.

    True or false?

    Answer:

  2. 10.The two triangles are congruent; sides with the same marks are equal. Which is written correctly?

    • A. △ABC ≅ △RQP
    • B. △ABC ≅ △QPR
    • C. △ABC ≅ △PQR
    • D. △ABC ≅ △PRQ

    Choose the right answer

    Answer:

  3. 11.In △ABC and △PQR, AB = PQ, ∠A = ∠P and ∠B = ∠Q. Which condition makes them congruent?

    • A. SAS
    • B. ASA
    • C. AAS
    • D. SSS

    Choose the right answer

    Answer:

  4. 12.In △ABC, ∠A = 34°, ∠B = 64° and BC = 5 cm; in △PQR, ∠P = 34°, ∠R = 64° and QR = 5 cm. The two triangles must be congruent.

    True or false?

    Answer:

Part C: Word problems (4 marks each)

  1. 13.△ABC is translated to △PQR (A to P, B to Q, C to R). AB = 15 cm, ∠B = 59° and ∠C = 46°. PQ = cm, ∠P = °.

    Fill in each blank

    Working:

    Answer:

  2. 14.A pole AD stands upright on the ground BC, and two equal cables AB and AC run from its top to the ground. In △ABD and △ACD, ∠ADB = ∠ADC = 90°, AB = AC and AD = AD. Which condition makes them congruent?

    • A. SAS
    • B. SSS
    • C. ASA
    • D. RHS

    Choose the right answer

    Working:

    Answer:

  3. 15.One angle of an isosceles triangle is 150°. What are the other two?

    • A. both 30°
    • B. both 150°
    • C. both 90°
    • D. both 15°

    Choose the right answer

    Working:

    Answer:

  4. 16.In △ABC, ∠A = 112° and ∠B = 34°. Which two sides must be equal?

    • A. AB = BC
    • B. AB = AC
    • C. No two sides are equal
    • D. AC = BC

    Choose the right answer

    Working:

    Answer:

  5. 17.Two fountains A and B in a park are 59 m apart. A lamp P is put on the perpendicular bisector of AB, 37 m from A. How far is P from B?

    Working:

    Answer:

  6. 18.Bisecting a 90° angle 3 times gives °; bisecting a 60° angle 2 times gives °.

    Fill in each blank

    Working:

    Answer:

Show the answers and working

Congruent triangles worksheet: Answers and working

Part A: Working out and filling in

  1. 1. 12, 36, 100

    The reflection keeps it congruent: A′B′ = 4 cm and ∠B′ = 100°. Enlarged 3 times, every side becomes 3 times as long: A″B″ = 3 × 4 = 12 cm, and the perimeter = 3 × 12 = 36 cm; but the shape stays the same, so the angles don't grow: ∠B″ = 100°. So △A″B″C″ is not congruent to △ABC.

  2. 2. 9, 6

    Sides with the same marks correspond and are equal (corr. sides, ≅△s): x = 9, y = 6.

  3. 3. B → C → A

    △ABC ≅ △ADC (SSS) first; only then do the corresponding angles give ∠DAC = ∠BAC = 38°, and last ∠BAD = 76°.

  4. 4. 34, 106

    △ABC ≅ △ADC (SSS), so ∠BAC = ∠DAC = 68° ÷ 2 = 34° and ∠BCA = 80° ÷ 2 = 40° (corr. ∠s, ≅△s); ∠ABC = 180° − 34° − 40° = 106°.

  5. 5. 52 cm

    The height to the base is also its median (prop. of isos. △): DC = BD = 11 cm, BC = 22 cm; perimeter = 15 + 15 + 22 = 52 cm.

  6. 6. 49, 6

    ∠ABC = 98° − 49° = 49° (ext. ∠ of △), so ∠ABC = ∠BAC and BC = AC = 6 cm (sides opp. equal ∠s).

  7. 7. C → B → A

    P above AB first, then Q below, then join PQ; PQ bisects AB at right angles, AM = MB = 8 ÷ 2 = 4 cm.

  8. 8. 45

    60° bisected 2 times is 15°, 60° bisected 1 times is 30°, and together 15° + 30° = 45°.

Part B: Choose, or true or false

  1. 9. False

    A rotation keeps the size, so the image's sides must stay 5 cm.

  2. 10. C. △ABC ≅ △PQR

    Sides with the same marks correspond, and so do the vertices they face; in a ≅ statement, corresponding vertices go in the same places.

  3. 11. B. ASA

    The side between ∠A and ∠B is AB itself: two angles and the side between, ASA.

  4. 12. True

    Find the third angles first: ∠C = 180° − 34° − 64° = 82° and ∠Q = 180° − 34° − 64° = 82° (∠ sum of △). Only when the two triangles have the same three angles, with BC and QR both facing the 34° angle, does AAS prove them congruent; with different angles they can't be.

Part C: Word problems

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

  1. 13. 15, 75

    The translated image is congruent: PQ = AB = 15 cm, ∠P = ∠A = 180° − 59° − 46° = 75°.

  2. 14. D. RHS

    Both are right-angled, the hypotenuses AB = AC, and AD is a common side: RHS.

  3. 15. D. both 15°

    150° is obtuse, so it must be the top angle; each base angle = (180° − 150°) ÷ 2 = 15°.

  4. 16. B. AB = AC

    ∠C = 180° − 112° − 34°. The two equal angles face the two equal sides (sides opp. equal ∠s).

  5. 17. 37 m

    Every point on the perpendicular bisector is as far from A as from B: PB = PA = 37 m.

  6. 18. 11.25, 15

    Each bisection halves the angle: 90° bisected 3 times is 11.25°, 60° bisected 2 times is 15°.

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