Question

reasoning about definitions and properties of geometric figures
determine if each statement is always, sometimes, or never true.

1. parallel lines are
2. perpendicular lines are coplanar.
3. distance around an unmarked circle can be measured.

Explanation:

Question 1: Parallel lines are $\boldsymbol{

ule{1cm}{0.15mm}}$ coplanar.

Step 1: Recall the definition of parallel lines.

Parallel lines are lines in a plane (or in space) that never meet and are always the same distance apart. By definition, in the context of Euclidean geometry, parallel lines lie in the same plane (coplanar). If lines are not coplanar, they are skew lines, not parallel.

Step 2: Determine the truth value.

Since the definition of parallel lines (in the standard geometric sense) requires them to be in the same plane (coplanar), this statement is always true.

Step 1: Recall the definition of perpendicular lines.

Perpendicular lines are lines that intersect at a right angle (90 degrees). For two lines to intersect, they must lie in the same plane (coplanar). If two lines are not coplanar, they cannot intersect, so they can't be perpendicular (as perpendicularity requires intersection at 90 degrees).

Step 2: Determine the truth value.

Since perpendicular lines must intersect (at 90 degrees), and intersecting lines are coplanar, this statement is always true.

Step 1: Recall the concept of the circumference of a circle.

The distance around a circle is its circumference, given by the formula $C = 2\pi r$ or $C=\pi d$, where $r$ is the radius and $d$ is the diameter. Even if a circle is unmarked (no given radius/diameter), we can measure the circumference using methods like wrapping a string around the circle and then measuring the length of the string, or using geometric principles (e.g., if we can find the radius/diameter through other means, like using similar figures or indirect measurement).

Step 2: Determine the truth value.

Since there are methods (direct or indirect) to measure the circumference (distance around) of an unmarked circle, this statement is always true (or sometimes? Wait, no—actually, in theory, we can always measure it with appropriate tools/methods. Wait, maybe "sometimes" is incorrect. Wait, let's re-examine. If the circle is, say, a theoretical circle with no physical presence, but the problem says "distance around an unmarked circle"—assuming it's a physical circle, we can measure it. So the answer is "always" or "sometimes"? Wait, no—actually, the key is: can we measure it? Yes, by using a measuring tape, string, etc. So the statement "Distance around an unmarked circle can be measured"—the word "can" implies that it is possible, which it is (using appropriate methods). So it is always true that it can be measured (i.e., the possibility exists). Wait, maybe the intended answer is "always" or "sometimes"? Wait, no—let's think again. The distance around a circle is its circumference. Even if unmarked, we can measure it (e.g., by rolling the circle and measuring the distance, or using a string). So the statement "can be measured" is always true (because the ability to measure it exists). So:

Step 2: Determine the truth value.

Since there are valid methods to measure the circumference of an unmarked circle (e.g., using a flexible measuring tool or geometric techniques), the distance around an unmarked circle can always be measured (in the sense that the action of measuring is possible). Thus, the statement is always true.

Answer:

always

Question 2: Perpendicular lines are $\boldsymbol{

ule{1cm}{0.15mm}}$ coplanar.