To show that two triangles are congruent, it is sufficient for one of the following three congruence conditions to hold.
- Three pairs of corresponding sides are equal (SSS)
- Two pairs of corresponding sides and the included angles are equal (SAS)
- One pair of corresponding sides and the angles at both ends are equal (ASA)
Example ①
In △ABC and △DEF, AB=DE=5cm, BC=EF=7cm, and CA=FD=6cm. Determine whether they are congruent.
Answer: Since three pairs of corresponding sides are equal, △ABC ≅ △DEF (SSS)
In △ABC and △DEF, AB=DE=5cm, BC=EF=7cm, and CA=FD=6cm. Determine whether they are congruent.
Answer: Since three pairs of corresponding sides are equal, △ABC ≅ △DEF (SSS)
Example ②
When AB=DE, ∠B=∠E, and BC=EF, which congruence condition can be used?
Answer: Two sides (AB=DE, BC=EF) and the included angle (∠B=∠E) → SAS
When AB=DE, ∠B=∠E, and BC=EF, which congruence condition can be used?
Answer: Two sides (AB=DE, BC=EF) and the included angle (∠B=∠E) → SAS
Key Points
- For SAS, it is important that the angle is the included angle (between the two sides, not at an outer end)
- For ASA, the angles are at both ends of the side
- Note that SSA and AAA are not congruence conditions
Practice Problems
- In △ABC and △DEF, AB=DE=4cm, ∠A=∠D=60°, and AC=DF=6cm. State the congruence condition.
- In △ABC and △DEF, ∠A=∠D=50°, AB=DE=5cm, and ∠B=∠E=70°. State the congruence condition.
Answer and Explanation
- Answer: SAS (AB=DE, ∠A=∠D, AC=DF: two sides and the included angle).
- Answer: ASA (∠A=∠D, AB=DE, ∠B=∠E: one side and the angles at both ends).