Yes, two equilateral triangles can always be congruent. This is because the definition of an equilateral triangle includes that all three sides are of equal length and all three angles are equal. Therefore, if two triangles have the same side lengths and the same angle measures, they are congruent by definition.

In other words, if two triangles are equilateral, it implies that they have the same size and shape, making them congruent to each other. So, any two equilateral triangles will be congruent to each other.

In an isosceles triangle, two sides are of equal length. Let's denote the length of these two equal sides as \(a\) cm. The third side is referred to as the base.

Given that the lengths of two sides of the isosceles triangle are 5 cm and 8 cm, we can set up the equation:

\[a = 5 \text{ cm}\]

Now, we know that the perimeter of a triangle is the sum of the lengths of all its sides. So, the perimeter (\(P\)) of the isosceles triangle can be calculated as:

\[P = a + a + \text{base}\]

Substituting the known values, we get:

\[P = 5 \text{ cm} + 5 \text{ cm} + 8 \text{ cm}\]
\[P = 10 \text{ cm} + 8 \text{ cm}\]
\[P = 18 \text{ cm}\]

So, the perimeter of the isosceles triangle is 18 cm.

To prove that the lengths of altitudes drawn to equal sides of an isosceles triangle are also equal:

(i) Given that in triangle PQR, PR = PQ (isosceles triangle), and altitudes QT and RS are drawn to equal sides PQ and PR respectively.

We'll prove that angle TRQ is equal to angle SQR.

Given: PR = PQ (isosceles triangle)

Prove: ∠TRQ = ∠SQR

Proof:

 Since QT and RS are altitudes, they are perpendicular to their respective bases.
∠QTR = 90° and ∠SRP = 90°

Since triangle PQR is isosceles, PR = PQ, so base angles ∠PQR = ∠PRQ.

In △QTR and △SRP:
∠TRQ = ∠PQR (Alternate Interior Angles)
∠SQR = ∠PRQ (Alternate Interior Angles)

Since ∠PQR = ∠PRQ (base angles of an isosceles triangle are equal), 
∠TRQ = ∠SQR

Thus, we have proved that angle TRQ is equal to angle SQR.

(ii) If ∠TRQ = 30°, then using the fact that ∠TRQ = ∠SQR (as proved above), ∠SQR = 30°.

Since the sum of angles in a triangle is 180°, and we know that ∠SQR = ∠SQR = 30°, we can find the measure of the base angles:

 ∠PQR = ∠PRQ = (180° - 30° - 30°) / 2 = (180° - 60°) / 2 = 120° / 2 = 60°