Congruent triangles · Ages 13–14
Proving triangles congruent
One pair a line, with its reason; look for common sides, opposite angles, parallels.
How to do it
Method
One pair a line, with its reason; look for common sides, opposite angles, parallels
AB = AC (given), BD = CD (given), AD = AD (common side), △ABD ≅ △ACD (SSS)
Another way
Work back from the end
To prove BD = CD: look at △ABD and △ACD
In full
How a proof is written: start "In △ABD and △ACD,", then a line for each pair of equal parts with its reason in brackets, and last the conclusion with its condition. AB = AC (given); BD = CD (given); AD = AD (common side); so △ABD ≅ △ACD (SSS). A question won't hand you every fact: look for them yourself — a side the two triangles share (a common side), vertically opposite angles where two lines cross, alternate or corresponding angles of parallel lines. Once the triangles are congruent, their corresponding parts prove more: ∠BAD = ∠CAD (corr. ∠s, ≅△s). Another way: work back from what you must prove. To prove BD = CD, find two triangles containing BD and CD, then see which facts they need.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
In ABCD, AB // DC and AD // BC, so △ABC ≅ △CDA (ASA). In the picture AB = 6 cm and AD = 4 cm. CD =
Show the answer and working
Answer: 6 cm
△ABC ≅ △CDA: A goes with C and B with D, so AB goes with CD: CD = AB = 6 cm (corr. sides, ≅△s).
Example 2
In ABCD, AB // DC and AD // BC. In △ABC and △CDA: ∠BAC = ∠DCA (alt. ∠s, AB // DC), AC = CA (common side), ∠BCA = ∠DAC (alt. ∠s, AD // BC), so △ABC ≅ △CDA ( ? ). Which condition goes in the gap?
- A. AAS
- B. ASA
- C. SSS
- D. SAS
Choose the right answer
Show the answer and working
Answer: B. ASA
In △ABC, AC lies between ∠BAC and ∠BCA: two angles and the side between, ASA.
Example 3
In the picture, AB = AD, CB = CD, ∠BAD = 70° and ∠BCD = 120°. ∠BAC = °, ∠ABC = °.
Fill in each blank
Show the answer and working
Answer: 35, 85
△ABC ≅ △ADC (SSS), so ∠BAC = ∠DAC = 70° ÷ 2 = 35° and ∠BCA = 120° ÷ 2 = 60° (corr. ∠s, ≅△s); ∠ABC = 180° − 35° − 60° = 85°.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.AD and BC cross at E, the midpoint of both. In △ABE and △DCE: AE = DE (given), ∠AEB = ∠DEC ( ? ), BE = CE (given), so △ABE ≅ △DCE (SAS). Which reason goes in the gap?
- A. common side
- B. vert. opp. ∠s
- C. given
- D. alt. ∠s, AB // DC
Choose the right answer
Answer:
2.In ABCD, AB // DC and AD // BC, so △ABC ≅ △CDA (ASA). In the picture AB = 6 cm and AD = 3 cm. CD =
Answer:
3.In ABCD, AB // DC and AD // BC. In △ABC and △CDA: ∠BAC = ∠DCA (alt. ∠s, AB // DC), AC = CA (common side), ∠BCA = ∠DAC (alt. ∠s, AD // BC), so △ABC ≅ △CDA ( ? ). Which condition goes in the gap?
- A. ASA
- B. SAS
- C. AAS
- D. SSS
Choose the right answer
Answer:
4.In the picture, AB = AD, CB = CD, ∠BAD = 64° and ∠BCD = 116°. ∠BAC = °, ∠ABC = °.
Fill in each blank
Answer:
5.In the picture, AB = AD and CB = CD. x =
Answer:
All the answers
- 1. B. vert. opp. ∠s
- 2. 6 cm
- 3. A. ASA
- 4. 32, 90
- 5. 95°
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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