Can a triangle be constructed with side lengths PQ = 3.3 cm, QR = 4.6 cm, and PR = 8.2 cm?

Given side lengths:
$PQ = 3.3 \text{ cm}$
$QR = 4.6 \text{ cm}$
$PR = 8.2 \text{ cm}$

Step 1: Check the Triangle Inequality Theorem.
We need to check if $PQ + QR > PR$, $PQ + PR > QR$, and $QR + PR > PQ$.

• Check 1: $PQ + QR > PR$ $3.3 \text{ cm} + 4.6 \text{ cm} = 7.9 \text{ cm}$ Is $7.9 \text{ cm} > 8.2 \text{ cm}$? No, $7.9 \text{ cm}$ is not greater than $8.2 \text{ cm}$.

Since the first condition of the Triangle Inequality Theorem is not met ($7.9 \text{ cm} \ngtr 8.2 \text{ cm}$), a triangle with these side lengths cannot be constructed.

Therefore, it is impossible to construct $\triangle PQR$ with the given side lengths. Consequently, the angles of such a triangle cannot be measured.

The triangle cannot be constructed because the sum of the lengths of sides $PQ$ and $QR$ is less than the length of side $PR$.