What is Congruence?

Congruent figures are shapes that have exactly the same size and shape. When you place one congruent figure on top of the other, they match perfectly.

Key property: All corresponding sides are equal in length, and all corresponding angles are equal in measure.

We write "triangle ABC is congruent to triangle PQR" as:

$\Delta ABC \cong \Delta PQR$

The symbol "≅" means "congruent to."

Congruence of Triangles

To prove two triangles are congruent, we need only three pieces of information. These are called postulates:

1. Side-Side-Side (SSS) Postulate

If three sides of one triangle equal three sides of another triangle, the triangles are congruent.

Example: If AB = PQ, BC = QR, and AC = PR, then ΔABC ≅ ΔPQR.

2. Side-Angle-Side (SAS) Postulate

If two sides and the included angle of one triangle equal two sides and the included angle of another triangle, the triangles are congruent.

3. Angle-Angle-Side (AAS) Postulate

If two angles and one side of one triangle equal two angles and the corresponding side of another triangle, the triangles are congruent.

4. Angle-Side-Angle (ASA) Postulate

If two angles and the included side of one triangle equal two angles and the included side of another triangle, the triangles are congruent.

5. Right angle-Hypotenuse-Side (RHS) Postulate

For right-angled triangles: if the hypotenuse and one side of one triangle equal the hypotenuse and one side of another triangle, they are congruent.

Worked Example:

In triangle ABC, AB = AC (an isosceles triangle). D is the midpoint of BC. Prove that ΔABD ≅ ΔACD.

Solution:

• AB = AC (given, isosceles triangle)
• BD = DC (D is midpoint)
• AD is common to both triangles

By SSS postulate: ΔABD ≅ ΔACD

Therefore, all corresponding angles are equal.

What is Similarity?

Similar figures have the same shape but different sizes. They maintain the same ratio of corresponding sides and have equal corresponding angles.

Key property: Corresponding angles are equal, and corresponding sides are proportional.

We write "triangle ABC is similar to triangle PQR" as:

$\Delta ABC \sim \Delta PQR$

The ratio of corresponding sides is called the constant of proportionality or scale factor.

Similar Triangles

Two triangles are similar if any of these conditions are met:

1. Angle-Angle (AA) Theorem

If two pairs of corresponding angles are equal, the triangles are similar.

Example: If ∠A = ∠P and ∠B = ∠Q, then ΔABC ∼ ΔPQR (the third angles are also equal).

2. Side-Side-Side (SSS) Similarity Theorem

If three pairs of corresponding sides are proportional, the triangles are similar.

Example: If AB/PQ = BC/QR = AC/PR, then ΔABC ∼ ΔPQR.

3. Side-Angle-Side (SAS) Similarity Theorem

If two pairs of corresponding sides are proportional and the included angles are equal, the triangles are similar.

Worked Example:

In the figure below, DE is parallel to AB. Given CE = 9 cm, AE = 12 cm, and DE = 6 cm. Find AB.

Solution:

Since DE ∥ AB:

• ∠CDE = ∠CAB (corresponding angles)
• ∠CED = ∠CBA (corresponding angles)

By AA similarity: ΔCDE ∼ ΔCAB

Therefore:

$\frac{CD}{CA} = \frac{DE}{AB}$

We know CE = 9 cm, AE = 12 cm, so CA = CE + AE = 21 cm.

$\frac{21 - CD}{21} = \frac{6}{AB}$

From the proportion of the triangle: CE/AE = DE/AB

$\frac{9}{12} = \frac{6}{AB}$

$AB = \frac{6 \times 12}{9} = 8 \text{ cm}$

So AB = 8 cm.

In Tanzania, congruence and similarity are used in construction and design. For example, when a mason at a building site in Dar es Salaam needs to cut bricks for a patterned wall, they use congruent shapes to ensure each piece fits perfectly. Similarly, architects use similarity to create scale drawings of buildings—like drawing a plan where 1 centimeter represents 1 meter—so they can design structures before actual construction begins. This helps ensure proportions are correct even when the actual building is much larger than the drawing.