Question

Complete each statement with the word always, sometimes, or never.

3. If $\mathbf{\text{a}}\mathbf{\perp }\mathbf{\text{c}}$ and $\mathbf{\text{b}}\mathbf{\perp }\mathbf{\text{c}}$, then a and b are _______ parallel.

Short answer

The sentence is:

If $\mathbf{\text{a}}\mathbf{\perp }\mathbf{\text{c}}$and $\mathbf{\text{b}}\mathbf{\perp }\mathbf{\text{c}}$, then a and b are sometimes parallel.

Step-by-step solution

Step 1. Define parallel lines and perpendicular lines.

Parallel lines are lines in a plane that are the same distance apart always.

Perpendicular lines are the lines that cross at $90°$.

Step 2. State the explanation.

Consider the case, skew lines a and b, they are both perpendicular to c, but not parallel.

So, if $a\perp c$and $b\perp c$ , then a and b are sometimes parallel.

Here, the correct word is sometimes.

Step 3. State the conclusion.

Therefore, the statement is:

If $a\perp c$and $b\perp c$, then a and b are sometimes parallel.