Shapes and symmetry
Recognising congruent shapes
Identical in shape and size — even if one has been turned round or flipped over. Orientation does not count.
What congruent means
Shapes that are IDENTICAL IN SHAPE AND SIZE are CONGRUENT. Crucially, they can be pointing in different directions and still be congruent. A different ORIENTATION does not change the shape or the size.
Corresponding sides and angles
Take two congruent triangles LMN and XYZ. Side LM equals side XY — these are CORRESPONDING SIDES. Angle MLN equals angle YXZ — these are CORRESPONDING ANGLES. In congruent shapes, all corresponding sides are equal and all corresponding angles are equal: LM = XY, MN = YZ, LN = XZ ∠MLN = ∠YXZ, ∠LNM = ∠XZY, ∠NML = ∠ZYX
Naming angles with three letters
When an angle is written with three letters, the MIDDLE letter tells you which vertex the angle is at. So angle LMN is the angle at vertex M. Angle MLN is the angle at vertex L. ∠ is the mathematical symbol for 'angle'.
Using congruence to find missing values
If two triangles are congruent, matching up the corresponding parts tells you every missing length and angle. Given one triangle with sides 5.1 cm, 8.2 cm and 10.9 cm and angles 108°, 45° and 27° — the congruent triangle has exactly the same set. You just need to work out which part matches which.
Worked examples
What does congruent mean?
- Identical in shape AND size.
- Orientation does not matter.
Answer: Same shape, same size — turned or flipped is still congruent
Triangles LMN and XYZ are congruent — which side matches LM?
- Match the letters in order.
- L goes with X and M with Y.
Answer: XY
Which angle matches ∠MLN?
- The middle letter gives the vertex: L.
- L corresponds to X.
Answer: ∠YXZ
Where is angle LMN?
- The MIDDLE letter names the vertex.
Answer: At vertex M
One triangle has sides 5.1, 8.2 and 10.9 cm — what about its congruent twin?
- Congruent shapes have all corresponding sides equal.
Answer: Exactly the same three lengths
Does turning a shape change whether it is congruent?
- A different orientation changes neither shape nor size.
Answer: No — orientation does not count
Where you see this in real life
- Every tile in a floor pattern is congruent to the others, which is why they fit no matter how you rotate them.
- Mass-produced parts must be congruent — a replacement will not fit if it is even slightly the wrong size.
Key words
- Congruent
- identical in shape and size
- Corresponding sides
- sides in the same position in two congruent shapes
- Corresponding angles
- angles in the same position in two congruent shapes
- Orientation
- the direction a shape is facing
- Vertex
- a corner of a shape
- Identical
- exactly the same
- Angle notation
- ∠LMN means the angle at vertex M