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Geometry and proof · Ages 14–15

Proofs with quadrilaterals worksheet

Part of: Quadrilaterals

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Proofs with quadrilaterals worksheet

Geometry and proof · Ages 14–15

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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.In the picture, AB // DC and AB = DC. Prove AD // BC: put the steps in order.

    • A. ∠BAC = ∠DCA (alt. ∠s, AB // DC)
    • B. ∠BCA = ∠DAC (corr. ∠s, ≅△s), so AD // BC (alt. ∠s equal)
    • C. AB = CD (given), AC = CA (common side), so △ABC ≅ △CDA (SAS)

    Write the letters in the right order

    Answer:

  2. 2.In the parallelogram ABCD, the bisector of ∠DAB meets DC at F, with AD = 6 cm and AB = 9 cm. FC =

    Answer:

  3. 3.ABCD is a parallelogram whose diagonals cross at E. Prove AE = CE: put the steps in order.

    • A. AE = CE (corr. sides, ≅△s)
    • B. AB = CD (opp. sides of // gram), so △ABE ≅ △CDE (ASA)
    • C. ∠EAB = ∠ECD, ∠EBA = ∠EDC (alt. ∠s, AB // DC)

    Write the letters in the right order

    Answer:

  4. 4.In the parallelogram ABCD, the bisector of ∠DAB meets DC at F, with AD = 3 cm and AB = 8 cm. FC =

    Answer:

  5. 5.In the parallelogram ABCD, the bisector of ∠DAB meets DC at F, with AD = 3 cm and AB = 6 cm. FC =

    Answer:

  6. 6.In the parallelogram ABCD, the bisector of ∠DAB meets DC at F, with AD = 4 cm and AB = 5 cm. FC =

    Answer:

  7. 7.In the parallelogram ABCD, the bisector of ∠DAB meets DC at F, with AD = 7 cm and AB = 13 cm. FC =

    Answer:

  8. 8.In the parallelogram ABCD, the bisector of ∠DAB meets DC at F, with AD = 3 cm and AB = 10 cm. FC =

    Answer:

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

  1. 9.ABCD is a parallelogram. One line of a proof is: ∠BAC = ∠DCA ( ? ). Which reason goes in the gap?

    • A. vert. opp. ∠s
    • B. alt. ∠s, AD // BC
    • C. alt. ∠s, AB // DC
    • D. opp. ∠s of // gram

    Choose the right answer

    Answer:

  2. 10.ABCD is a parallelogram. In △ABC and △CDA: AB = CD ( ? ), … Which reason goes in the gap?

    • A. corr. sides, ≅△s
    • B. given
    • C. opp. sides of // gram
    • D. common side

    Choose the right answer

    Answer:

  3. 11.The parallelogram ABCD's diagonals cross at E. In △ABE and △CDE: ∠EAB = ∠ECD, AB = CD and ∠EBA = ∠EDC. So △ABE ≅ △CDE ( ? ). Which condition goes in the gap?

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

    Choose the right answer

    Answer:

  4. 12.ABCD is a parallelogram whose diagonals cross at E. One line of a proof is: AE = CE ( ? ). Which reason goes in the gap?

    • A. given
    • B. prop. of rhombus
    • C. diags. of // gram
    • D. corr. sides, ≅△s

    Choose the right answer

    Answer:

Part C: Word problems (4 marks each)

  1. 13.A parallelogram table top ABCD has AD = 6 dm, AB = 11 dm and ∠DAB = 58°. The bisector of ∠DAB meets DC at F. ∠DFA = °, DF = dm, FC = dm.

    Fill in each blank

    Working:

    Answer:

  2. 14.In ABCD, AB = CD = 12 cm and AD = BC = 3 cm. Prove AB // DC: put the steps in order.

    • A. AB = CD, BC = DA (given), AC = CA (common side)
    • B. ∠BAC = ∠DCA are alternate angles, so AB // DC (alt. ∠s equal)
    • C. So △ABC ≅ △CDA (SSS), ∠BAC = ∠DCA (corr. ∠s, ≅△s)

    Write the letters in the right order

    Working:

    Answer:

  3. 15.E and F lie on the diagonal AC of the parallelogram ABCD with AE = CF = 12 cm. Prove BE = DF: put the steps in order.

    • A. AE = CF = 12 cm (given), so △ABE ≅ △CDF (SAS)
    • B. BE = DF (corr. sides, ≅△s)
    • C. AB = CD (opp. sides of // gram), ∠BAE = ∠DCF (alt. ∠s, AB // DC)

    Write the letters in the right order

    Working:

    Answer:

  4. 16.The rhombus ABCD has sides of 9 cm and its diagonals cross at E. Proving AC ⊥ BD, one line is: AB = AD ( ? ), BE = DE (diags. of // gram), AE = AE (common side). Which reason goes in the gap?

    • A. corr. sides, ≅△s
    • B. common side
    • C. opp. sides of // gram
    • D. definition of rhombus (four equal sides)

    Choose the right answer

    Working:

    Answer:

  5. 17.In ABCD, AB = CD = 6 cm and AD = BC = 3 cm. Prove AB // DC: put the steps in order.

    • A. So △ABC ≅ △CDA (SSS), ∠BAC = ∠DCA (corr. ∠s, ≅△s)
    • B. AB = CD, BC = DA (given), AC = CA (common side)
    • C. ∠BAC = ∠DCA are alternate angles, so AB // DC (alt. ∠s equal)

    Write the letters in the right order

    Working:

    Answer:

  6. 18.The rhombus ABCD has sides of 5 cm and its diagonals cross at E. Proving AC ⊥ BD, one line is: AB = AD ( ? ), BE = DE (diags. of // gram), AE = AE (common side). Which reason goes in the gap?

    • A. opp. sides of // gram
    • B. corr. sides, ≅△s
    • C. common side
    • D. definition of rhombus (four equal sides)

    Choose the right answer

    Working:

    Answer:

Show the answers and working

Proofs with quadrilaterals worksheet: Answers and working

Part A: Working out and filling in

  1. 1. A → C → B

    First the angle between the two sides for congruence (SAS), then the corresponding angles give equal alternate angles.

  2. 2. 3 cm

    ∠DFA = ∠FAB (alt. ∠s, AB // DC) = ∠DAF, so DF = DA = 6 cm (sides opp. equal ∠s); DC = AB = 9 cm, so FC = 9 − 6 = 3 cm.

  3. 3. C → B → A

    Two pairs of alternate angles from the parallels, then the equal opposite sides for congruence (ASA), and last the corresponding sides.

  4. 4. 5 cm

    ∠DFA = ∠FAB (alt. ∠s, AB // DC) = ∠DAF, so DF = DA = 3 cm (sides opp. equal ∠s); DC = AB = 8 cm, so FC = 8 − 3 = 5 cm.

  5. 5. 3 cm

    ∠DFA = ∠FAB (alt. ∠s, AB // DC) = ∠DAF, so DF = DA = 3 cm (sides opp. equal ∠s); DC = AB = 6 cm, so FC = 6 − 3 = 3 cm.

  6. 6. 1 cm

    ∠DFA = ∠FAB (alt. ∠s, AB // DC) = ∠DAF, so DF = DA = 4 cm (sides opp. equal ∠s); DC = AB = 5 cm, so FC = 5 − 4 = 1 cm.

  7. 7. 6 cm

    ∠DFA = ∠FAB (alt. ∠s, AB // DC) = ∠DAF, so DF = DA = 7 cm (sides opp. equal ∠s); DC = AB = 13 cm, so FC = 13 − 7 = 6 cm.

  8. 8. 7 cm

    ∠DFA = ∠FAB (alt. ∠s, AB // DC) = ∠DAF, so DF = DA = 3 cm (sides opp. equal ∠s); DC = AB = 10 cm, so FC = 10 − 3 = 7 cm.

Part B: Choose, or true or false

  1. 9. C. alt. ∠s, AB // DC

    AC cuts the parallels AB and DC, and ∠BAC and ∠DCA are on either side of it: alt. ∠s, AB // DC.

  2. 10. C. opp. sides of // gram

    AB and CD are a pair of the parallelogram's opposite sides: opp. sides of // gram.

  3. 11. D. ASA

    AB lies between ∠EAB and ∠EBA: two angles and the side between, ASA.

  4. 12. C. diags. of // gram

    A parallelogram's diagonals bisect each other: diags. of // gram.

Part C: Word problems

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

  1. 13. 29, 6, 5

    ∠DFA = ∠FAB = 29° (alt. ∠s) = ∠DAF, so DF = DA = 6 dm (sides opp. equal ∠s); FC = 11 − 6 = 5 dm.

  2. 14. A → C → B

    Join AC: three equal sides give congruence (SSS), the corresponding angles are alternate angles, and equal alternate angles make the lines parallel.

  3. 15. C → A → B

    Two sides and the angle between (AB = CD, ∠BAE = ∠DCF, AE = CF) for congruence first, then the corresponding sides.

  4. 16. D. definition of rhombus (four equal sides)

    A rhombus has four equal sides, so AB = AD; then △ABE ≅ △ADE (SSS) and ∠AEB = ∠AED = 90°.

  5. 17. B → A → C

    Join AC: three equal sides give congruence (SSS), the corresponding angles are alternate angles, and equal alternate angles make the lines parallel.

  6. 18. D. definition of rhombus (four equal sides)

    A rhombus has four equal sides, so AB = AD; then △ABE ≅ △ADE (SSS) and ∠AEB = ∠AED = 90°.

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