Congruent Triangles

1Summary 重點摘要

Learn what congruent triangles are, and four conditions for congruence.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: The meaning of congruenceCongruent triangles are two triangles identical in both shape and size; after translating, rotating or flipping one, it overlaps the other exactly, with all corresponding sides equal and all corresponding angles equal, written as ≅, for example triangle ABC ≅ triangle DEF. Why study congruence? It is the basic tool of geometric proof: proving two segments or angles equal is often achieved by proving two triangles congruent. A common mistake is concluding congruence from three equal angles alone; the sizes may differ, which is similarity only.
2. Step 2: SSS, three pairs of equal sidesSSS states that if the three pairs of corresponding sides of two triangles are equal, the triangles are congruent. For example, triangle ABC with sides 3, 4, 5 and triangle DEF with sides 3, 4, 5 are congruent. Why do three sides fix the shape? Three side lengths uniquely determine a triangle's size and shape, like a frame made of three rigid rods that cannot deform. A common mistake is mismatching the correspondence, pairing the longest side with a short one; list the sides in size order when matching.
3. Step 3: SAS, two sides and the included angleSAS states that if two pairs of corresponding sides are equal and the angle between them is also equal, the triangles are congruent. For example, AB = DE = 3, AC = DF = 4 and the included angles ∠A = ∠D = 60° give congruent triangles. Why must the angle be included? Without the angle lying between the two known sides, the combination cannot fix a unique triangle and two different shapes may be drawn. A common mistake is using a non-included angle with SAS, the most classic exam trap; confirm the angle sits between the two sides first.
4. Step 4: ASA, two angles and the included sideASA states that if two pairs of corresponding angles are equal and the side between them is also equal, the triangles are congruent. For example, ∠A = ∠D = 50°, ∠B = ∠E = 60° and the included side AB = DE = 5 give congruent triangles. Why is the included side important? Equal angles fix only the shape; one side is needed to fix the size, and the included side joins the two angles exactly. A common mistake is claiming congruence from two equal angles alone, omitting the equal included side; an incomplete reason loses marks.
5. Step 5: AAS, two angles and a non-included sideAAS states that if two pairs of corresponding angles are equal and one pair of non-included sides is equal, the triangles are congruent. Why does it work? The angles of a triangle sum to 180°, so knowing two angles gives the third, and AAS reduces to ASA with the same conclusion. For example, given ∠A = ∠D, ∠B = ∠E and the non-included sides AC = DF, compute ∠C = ∠F from the angle sum and then apply ASA. A common mistake is confusing AAS with SSA, where the equal side is not between the angles and congruence is not guaranteed.
6. Step 6: Conditions that cannot be usedTwo seemingly reasonable conditions must be remembered as invalid: SSA, two sides and a non-included angle, cannot prove congruence, because the same data can draw two different triangles; AAA, three equal angles, cannot either, since equal angles fix only the shape, giving similarity, and sizes may differ, for example right triangles with sides 3, 4, 5 and 6, 8, 10 have equal corresponding angles yet are not congruent. A common mistake is citing these two conditions as reasons in proofs, which markers judge wrong; use only SSS, SAS, ASA and AAS.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Congruent = same shape same size
• SSS: three sides equal
• SAS: two sides included angle
• ASA: two angles included side
• SSA, AAA are not conditions