Question

Determine whether each statement is sometimes, always, or never true.

If the volumes of two triangular prisms are the same, the prisms are congruent.

Answer 1

Sometimes true

Explanation:

Volume = ⅓(base area × height)

The product 'base area × height' can be equal if the base and height are congruent, but there are other possibilities too

Example:

Base area of the second one is double but the height is half.

Answer 2

Sometimes

Explanation:

If the volumes of two triangular prisms are the same, that means that:

`(1)/(3) B_1h_1=(1)/(3) B_2h_2` , where B_1 and h_1 belong to one prism and B_2 and h_2 belong to the second.

It is possible that if the volumes are the same, the prisms are congruent. That means that B_1 = B_2 and h_1 = h_2. However, this isn't always the case. Here's a counterexample:

PRISM 1: B = 4, h = 3 ⇒ V = (1/3) * 4 * 3 = 4

PRISM 2: B = 2, h = 6 ⇒ V = (1/3) * 2 * 6 = 4

Their volumes are the same, but their dimensions certainly aren't. So this statement is true only sometimes.

Hope this helps!