Congruence of triangles

The following are conditions for two triangles to be congruent:

• SSS (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent. Consider the example below showing two triangles with equal lengths of the corresponding sides.

Example 1

Prove that the two triangles (ΔABC and ΔBCD) below are congruent.

Solution

Another condition:

• SAS (Side-Angle-Side): This means that we have two triangles where we know two sides and the included angles are equal. For example: if the two sides and the included angle of one triangle are equal to corresponding sides and the included angle of the other triangle, we say that the two triangles are congruent.

• ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to the two angles and included side of another triangle we say that the two triangles are congruent. For example:

• AAS (Angle-Angle-Side): If two angles and non-included side of one triangle are equal to the corresponding angles and non-included side of the other triangle, then the two triangles are congruent. For example:

• HL (Hypotenuse-Leg): This is applicable only to a right-angled triangle. The longest side of a right-angled triangle is called the hypotenuse and the other two sides are legs. It means we have two right-angled triangles with the same length of hypotenuse and the same length for one of the other two legs. If the hypotenuse and one leg of one right-angled triangle are equal to a corresponding hypotenuse and one leg of the other right-angled triangle, the two triangles are congruent. For example:

Important note: Do not use AAA (Angle-Angle-Angle). This means we are given all three angles of a triangle but no sides. This is not enough information to decide whether the two triangles are congruent or not because the triangles can have the same angles but different sizes. See the illustration below:

The two triangles are not congruent. Without knowing at least one side, we can't be sure that the triangles are congruent.

Prove that the two triangles (ΔABC and ΔBCD) below are congruent.

Solution

We are given a rectangle (ABCD); (AC = BD); (AB = CD) (they are opposite sides of a rectangle) and a diagonal (BC).

Required to prove: ΔABC ≅ ΔBCD

Proof:

$AC = BD \quad \text{Opposite sides of a rectangle}$

$AB = CD \quad \text{Opposite sides of a rectangle}$

AC is common

$\therefore \Delta ABC \cong \Delta BCD \text{ by SSS}$

Isosceles Triangle Theorem

The figure below illustrates an example of an isosceles triangle:

An isosceles triangle has two congruent sides (opposite sides) and two congruent angles. The congruent angles are called base angles and the other angle is called the vertex angle. Angles A and B are base angles and angle C is the vertex angle.

The Base Angle Theorem

If two sides of a triangle are congruent, then the angles opposite to these sides are congruent.

Required to prove: angle A = angle B

Proof: Let S be the midpoint of AB and then join C and S.

Since S is the midpoint of AB, AS = BS. We are given that AC = BC. CS is common (i.e. reflexive property).

Therefore by SSS, ΔACS ≅ ΔBCS.

Therefore, the base angles i.e. angle CAS and angle CBS are equal (by the definition of congruence of triangles).

It states that if two angles of a triangle are congruent, then sides opposite those angles are congruent.

We are given that: angle B = angle C. Required to prove: AB = AC.

Proof: Bisect angle A by drawing a line perpendicular to BC at P.

Since AP bisects angle A, then ∠BAP = ∠CAP. We are given that ∠ABP = ∠ACP. AP is common (reflexive property).

Therefore, by AAS the triangles ABP and CAP are congruent. Hence AB = AC (definition of congruence of triangles).