Congruent triangles (Edexcel GCSE Maths): Revision Notes
Congruent triangles
What are congruent triangles?
Congruent triangles are triangles that are exactly the same shape and size. When triangles are congruent, all their corresponding sides and angles are equal. This means you could place one triangle exactly on top of the other and they would match perfectly.
Think of congruent triangles like identical twins - they look exactly the same in every way. If you could cut out one triangle and place it over the other, every part would line up perfectly.
To prove that two triangles are congruent, you need to show that one of four specific conditions is satisfied. You don't need to prove all sides and angles are equal - just one of these four conditions is enough.
The four congruence conditions
1. SSS (Side-Side-Side)
Definition: All three corresponding sides of the triangles are equal in length.
This condition means that if you can show the three sides of one triangle match the three sides of another triangle, then the triangles must be congruent. The angles will automatically be equal too.
Worked Example: SSS Congruence
Triangle 1 has sides 3cm, 2.4cm, and 1.8cm. Triangle 2 has sides 3cm, 2.4cm, and 1.8cm.
Since all three corresponding sides are equal:
- Side 1: 3cm = 3cm ✓
- Side 2: 2.4cm = 2.4cm ✓
- Side 3: 1.8cm = 1.8cm ✓
Therefore, these triangles are congruent by SSS.
2. AAS (Angle-Angle-Side)
Definition: Two angles and one corresponding side are equal.
When two angles of a triangle are known, the third angle is automatically determined (since angles in a triangle sum to ). If you also know one corresponding side is equal, the triangles must be congruent.
Worked Example: AAS Congruence
Both triangles have:
- Angle A = 40°
- Angle B = 95°
- Side opposite to the 40° angle = 3cm
Since two angles and a corresponding side are equal, the triangles are congruent by AAS.
3. SAS (Side-Angle-Side)
Definition: Two sides and the included angle between them are equal.
Critical Point: The angle must be the one that sits between the two known sides. This is called the included angle. If the angle is not between the two sides, this condition doesn't apply!
Worked Example: SAS Congruence
Both triangles have:
- Side 1 = 6m
- Side 2 = 8m
- Included angle between these sides = 100°
Since two sides and the included angle are equal, the triangles are congruent by SAS.
4. RHS (Right angle-Hypotenuse-Side)
Definition: Both triangles are right-angled, their hypotenuses are equal, and one other corresponding side is equal.
This condition only applies to right-angled triangles. The hypotenuse is always the longest side and sits opposite the right angle (90°).
Worked Example: RHS Congruence
Both triangles have:
- A right angle (90°)
- Hypotenuse = 5cm
- Another corresponding side = 2cm
Since both are right triangles with equal hypotenuses and one equal side, the triangles are congruent by RHS.
Proving congruence in exam questions
When asked to prove triangles are congruent, follow this systematic approach:
Step-by-Step Approach to Proving Congruence
- Identify which condition you think applies (SSS, AAS, SAS, or RHS)
- State which sides or angles are equal and explain why
- Name the congruence condition used
- Conclude that the triangles are congruent
Worked Example: Complete Proof
Looking at triangles ABD and BDC:
Step 1: Identify the condition I think RHS applies since both triangles appear to be right-angled.
Step 2: State equal measurements
- Both triangles share side BD, so BD is equal in both triangles
- Both triangles are right-angled (given)
- Both triangles have the same hypotenuse length of 6cm (given)
Step 3: Name the condition The triangles satisfy the RHS condition.
Step 4: Conclude Therefore, triangle ABD ≅ triangle BDC (congruent).
Common sides
Key Concept: Common Sides
If two triangles share a side, then that side is equal in both triangles. This is often extremely useful when proving congruence, as it gives you one equal measurement automatically.
For example, if triangles ABC and ACD both contain side AC, then AC is equal in both triangles by definition.
Exam tips
Essential Exam Strategies
- Always clearly state which condition you're using (SSS, AAS, SAS, or RHS)
- For SAS, make sure the angle is definitely the included angle between the two sides you've identified
- Look for common sides or common angles between triangles - these are "free" equal measurements
- Show your working step by step - explain why each measurement is equal
- Use the standard abbreviations (SSS, AAS, SAS, RHS) in your final conclusion
- Always write your conclusion clearly: "Therefore, triangle ABC ≅ triangle XYZ"
Key Points to Remember
Essential Takeaways
- Congruent triangles are identical in shape and size
- Only one of the four conditions (SSS, AAS, SAS, RHS) needs to be satisfied to prove congruence
- For SAS, the angle must be the included angle between the two sides
- RHS only applies to right-angled triangles
- Common sides between triangles are automatically equal and can help with proofs
- Always follow the four-step proof process: Identify → State → Name → Conclude