Congruence

What is Congruence?

Congruence means exact match in shape and size.

• Two shapes are congruent if you could place one over the other and they fit perfectly.
• They might be flipped, turned, or moved—but they’re still congruent as long as they haven’t been resized.

Note: Congruent shapes are not enlargements of each other. Same size is essential

How Do We Prove Two Shapes Are Congruent?

To confirm that two shapes are congruent:

Check if matching sides are equal in length

Check if matching angles are the same

The orientation doesn’t need to match—it might be reflected or rotated

Transformations that preserve congruence:

• Translation (slide)
• Rotation (turn)
• Reflection (flip)

Example: Shapes

Question: Which shapes are congruent to shape A?

Compare the sides and angles of each shape with A.

Shape C and shape D are congruent to A—they match in size and shape, even if the orientation is different.

Congruent Triangles

Triangles are congruent if they have the same size and shape.

They may be reflected, rotated, or moved.

But all three sides and three angles must be equal between them.

Tests for Triangle Congruence

You only need to match three elements—but they must be the right combination. There are five valid tests:

Not valid: AAA and SSA do not guarantee congruence.

Example: Triangles

Prove that triangle ABC is congruent to triangle PQR.

• $\angle \text{ABC} = \angle \text{RPQ} = 25\degree$
• $\angle \text{BAC} = \angle \text{PRQ} = 90\degree$
• Side $\text{AB}$ = Side $\text{PR}$ = 6 cm

Two angles and the side between them match $\to$ ASA condition applies.

$\therefore$ Triangle ABC ≅ Triangle PQR by ASA.