Geometry and proof · Ages 13–14

Proving triangles congruent 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, AB = AD, CB = CD and ∠BAC = 26°. Find ∠BAD: put the steps in order.
- A. ∠DAC = ∠BAC = 26° (corr. ∠s, ≅△s)
- B. AB = AD, CB = CD, AC = AC, so △ABC ≅ △ADC (SSS)
- C. ∠BAD = 26° + 26° = 52°
Write the letters in the right order
Answer:
2.In ABCD, AB // DC and AD // BC, so △ABC ≅ △CDA (ASA). In the picture AB = 5 cm and AD = 4 cm. CD =
Answer:
3.In ABCD, AB // DC and AD // BC, so △ABC ≅ △CDA (ASA). In the picture AB = 9 cm and AD = 6 cm. CD =
Answer:
4.In the picture, AB = AD, CB = CD and ∠BAC = 39°. Find ∠BAD: put the steps in order.
- A. ∠BAD = 39° + 39° = 78°
- B. ∠DAC = ∠BAC = 39° (corr. ∠s, ≅△s)
- C. AB = AD, CB = CD, AC = AC, so △ABC ≅ △ADC (SSS)
Write the letters in the right order
Answer:
5.In the picture, AB = AD, CB = CD and ∠BAC = 27°. Find ∠BAD: put the steps in order.
- A. ∠BAD = 27° + 27° = 54°
- B. AB = AD, CB = CD, AC = AC, so △ABC ≅ △ADC (SSS)
- C. ∠DAC = ∠BAC = 27° (corr. ∠s, ≅△s)
Write the letters in the right order
Answer:
6.In ABCD, AB // DC and AD // BC, so △ABC ≅ △CDA (ASA). In the picture AB = 5 cm and AD = 3 cm. CD =
Answer:
7.In the picture, AB = AD, CB = CD, ∠BAD = 78° and ∠BCD = 102°. ∠BAC = °, ∠ABC = °.
Fill in each blank
Answer:
8.In the picture, AB = AD and CB = CD. x =
Answer:
9.In the picture, AB = AD, CB = CD, ∠BAD = 86° and ∠BCD = 92°. ∠BAC = °, ∠ABC = °.
Fill in each blank
Answer:

13.AD and BC cross at E, AE = DE and BE = CE. Prove AB // DC: put the steps in order.
- A. ∠BAD = ∠CDA, so AB // DC (alt. ∠s equal)
- B. ∠BAE = ∠CDE (corr. ∠s, ≅△s)
- C. AE = DE, ∠AEB = ∠DEC (vert. opp. ∠s), BE = CE, so △ABE ≅ △DCE (SAS)
Write the letters in the right order
Working:
Answer:
14.A kite ABCD, as shown, has AB = AD, CB = CD, ∠BAD = 96° and ∠ABC = 66°. ∠ADC = °, ∠BCD = °.
Fill in each blank
Working:
Answer:
15.In △ABC and △ADC: AB = AD (given), ∠BAC = ∠DAC (given), AC = AC (common side), so △ABC ≅ △ADC ( ? ). Which condition goes in the gap?
- A. ASA
- B. SSS
- C. AAS
- D. SAS
Choose the right answer
Working:
Answer:
16.In ABCD, AB // DC and AD // BC. One line of the proof is: ∠BAC = ∠DCA ( ? ). Which reason goes in the gap?
- A. alt. ∠s, AD // BC
- B. corr. ∠s, ≅△s
- C. alt. ∠s, AB // DC
- D. vert. opp. ∠s
Choose the right answer
Working:
Answer:
17.In ABCD, AB // DC and AD // BC. Prove AB = DC: put the steps in order.
- A. AB = CD (corr. sides, ≅△s)
- B. AC = CA (common side), so △ABC ≅ △CDA (ASA)
- C. ∠BAC = ∠DCA, ∠BCA = ∠DAC (alt. ∠s)
Write the letters in the right order
Working:
Answer:
18.A kite ABCD, as shown, has AB = AD, CB = CD, ∠BAD = 90° and ∠ABC = 67°. ∠ADC = °, ∠BCD = °.
Fill in each blank
Working:
Answer:

Show the answers and working

Proving triangles congruent worksheet: Answers and working

1. B → A → C
△ABC ≅ △ADC (SSS) first; only then do the corresponding angles give ∠DAC = ∠BAC = 26°, and last ∠BAD = 52°.
2. 5 cm
△ABC ≅ △CDA: A goes with C and B with D, so AB goes with CD: CD = AB = 5 cm (corr. sides, ≅△s).
3. 9 cm
△ABC ≅ △CDA: A goes with C and B with D, so AB goes with CD: CD = AB = 9 cm (corr. sides, ≅△s).
4. C → B → A
△ABC ≅ △ADC (SSS) first; only then do the corresponding angles give ∠DAC = ∠BAC = 39°, and last ∠BAD = 78°.
5. B → C → A
△ABC ≅ △ADC (SSS) first; only then do the corresponding angles give ∠DAC = ∠BAC = 27°, and last ∠BAD = 54°.
6. 5 cm
△ABC ≅ △CDA: A goes with C and B with D, so AB goes with CD: CD = AB = 5 cm (corr. sides, ≅△s).
7. 39, 90
△ABC ≅ △ADC (SSS), so ∠BAC = ∠DAC = 78° ÷ 2 = 39° and ∠BCA = 102° ÷ 2 = 51° (corr. ∠s, ≅△s); ∠ABC = 180° − 39° − 51° = 90°.
8. 98°
AB = AD, CB = CD, AC = AC (common side), so △ABC ≅ △ADC (SSS): x = ∠ABC = 180° − 39° − 43° = 98° (corr. ∠s, ≅△s; ∠ sum of △).
9. 43, 91
△ABC ≅ △ADC (SSS), so ∠BAC = ∠DAC = 86° ÷ 2 = 43° and ∠BCA = 92° ÷ 2 = 46° (corr. ∠s, ≅△s); ∠ABC = 180° − 43° − 46° = 91°.

10. A. common side
AC is a side both triangles share (common side): three pairs of equal sides, so SSS.
11. C. vert. opp. ∠s
AD and BC cross at E, and ∠AEB and ∠DEC face each other: vertically opposite angles.
12. B. ASA
In △ABC, AC lies between ∠BAC and ∠BCA: two angles and the side between, ASA.

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

13. C → B → A
Congruence first (SAS), then the corresponding angles give equal alternate angles, and last AB // DC.
14. 66, 132
AC = AC (common side), so △ABC ≅ △ADC (SSS) and ∠ADC = ∠ABC = 66° (corr. ∠s, ≅△s); ∠BCD = 360° − 96° − 2 × 66° = 132° (∠ sum of polygon).
15. D. SAS
∠BAC lies between AB and AC: two sides and the angle between, SAS.
16. C. alt. ∠s, AB // DC
AC cuts the parallel lines AB and DC; ∠BAC and ∠DCA are on either side of it, between the lines: alt. ∠s, AB // DC.
17. C → B → A
Two pairs of alternate angles from the parallels first, then the common side for congruence (ASA), and last the corresponding sides.
18. 67, 136
AC = AC (common side), so △ABC ≅ △ADC (SSS) and ∠ADC = ∠ABC = 67° (corr. ∠s, ≅△s); ∠BCD = 360° − 90° − 2 × 67° = 136° (∠ sum of polygon).

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