Congruent triangles · Ages 13–14
Conditions for congruent triangles
SSS, SAS, ASA, AAS, RHS; never AAA or SSA.
How to do it
Method
SSS, SAS, ASA, AAS, RHS; never AAA or SSA
AB = PQ, ∠B = ∠Q, BC = QR: ∠B lies between the sides, SAS
Another way
Find the third angle first
Two angles equal, so the third is too: AAS becomes ASA
In full
You needn't measure all six parts: three suitable ones are enough. SSS, three sides equal; SAS, two sides and the angle between them (the angle must be between the two sides); ASA, two angles and the side between them; AAS, two angles and a side facing one of them; RHS, in two right-angled triangles, the hypotenuse and one other side. Not enough: AAA, the angles equal but the sizes free; SSA, two sides and an angle not between them, which can make two different triangles. The method: count the S's (sides) and A's (angles), then check whether the angle sits between the sides. Another way: two angles known means the third is known too (∠ sum of △), so AAS can also be seen as ASA.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
By the marks (parts with the same marks are equal; a square is a right angle), △ABC and △PQR must be congruent.
True or false?
Show the answer and working
Answer: True
SSS, SAS, ASA, AAS and RHS each prove congruence; three equal angles (AAA), or two sides and an angle not between them (SSA), are not enough: the triangles may differ.
Example 2
Two triangles have two pairs of equal angles, 51° and 62°. Their third angles are both
Show the answer and working
Answer: 67°
180° − 51° − 62° = 67° (∠ sum of △): two equal angles force the third to be equal, which is why AAS can be seen as ASA.
Example 3
In △ABC and △PQR, AB = PQ, ∠B = ∠Q and ∠C = ∠R. Which condition makes them congruent?
- A. AAS
- B. SSS
- C. SAS
- D. ASA
Choose the right answer
Show the answer and working
Answer: A. AAS
Two pairs of angles (∠B, ∠C) and a pair of sides not between them (AB faces ∠C): AAS.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.By the marks (parts with the same marks are equal; a square is a right angle), △ABC and △PQR must be congruent.
True or false?
Answer:
2.In the picture, AB = PQ and AC = PR. Which other fact proves △ABC ≅ △PQR by SAS?
- A. ∠A = ∠P
- B. ∠B = ∠Q
- C. ∠C = ∠R
- D. BC = QR
Choose the right answer
Answer:
3.Two triangles have two pairs of equal angles, 69° and 83°. Their third angles are both
Answer:
4.In △ABC, ∠A = 45°, ∠B = 53° and BC = 11 cm; in △PQR, ∠P = 45°, ∠R = 82° and QR = 11 cm. The two triangles must be congruent.
True or false?
Answer:
5.In the picture, AB = PQ and ∠B = ∠Q. Which one more fact would still not prove △ABC ≅ △PQR?
- A. ∠C = ∠R
- B. AC = PR
- C. ∠A = ∠P
- D. BC = QR
Choose the right answer
Answer:
All the answers
- 1. True
- 2. A. ∠A = ∠P
- 3. 28°
- 4. True
- 5. B. AC = PR
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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