Question

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

1. parallel lines are always coplanar.
2. perpendicular lines are never coplanar.
3. distance around an unmarked circle can be measured.

x always
sometimes
never

Explanation:

Problem 2:

Step1: Recall the definition of perpendicular lines

Perpendicular lines are lines that intersect at a right angle ($90^\circ$). By the definition of coplanar lines (lines that lie in the same plane), any two lines that intersect (including perpendicular lines) must lie in the same plane. So perpendicular lines are always coplanar. The original answer of "never" is incorrect.

Step2: Correct the statement

So the correct answer for "Perpendicular lines are ____ coplanar" is "always".

Problem 3:

Step1: Recall the concept of the distance around 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 (we don't know the radius or diameter initially), we can measure the circumference using methods like wrapping a string around the circle and then measuring the length of the string, or using other geometric or physical measurement techniques. So the distance around an unmarked circle can always be measured.

Step2: Choose the correct option

So the correct option is "always".

Corrected Answers:

1. Perpendicular lines are always coplanar.
2. Distance around an unmarked circle can always be measured.

Answer:

Step1: Recall the concept of the distance around 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 (we don't know the radius or diameter initially), we can measure the circumference using methods like wrapping a string around the circle and then measuring the length of the string, or using other geometric or physical measurement techniques. So the distance around an unmarked circle can always be measured.

Step2: Choose the correct option

So the correct option is "always".

Corrected Answers:

1. Perpendicular lines are always coplanar.
2. Distance around an unmarked circle can always be measured.