Question

write each biconditional as a conjunction of two conditionals. then state whether the biconditional is true or false.

1. point p is between a and b if and only if p, a, and b are collinear.
2. segments are congruent if and only if their lengths are equal.
3. two planes never intersect if and only if they are parallel.
4. point m is the midpoint of $overline{xy}$ if and only if $xm = my$.

write the inverse and contrapositive of each conditional.

1. if a polygon is a quadrilateral, then the polygon is not a pentagon.
2. if two angles have the same measure, then they are congruent.
3. if the computer program has no bugs, then it will work flawlessly.
4. if two lines intersect, then their intersection is exactly one point.

Explanation:

24.

Step1: Write as two conditionals

Conditional 1: If point $P$ is between $A$ and $B$, then $P$, $A$, and $B$ are collinear. Conditional 2: If $P$, $A$, and $B$ are collinear, then point $P$ is between $A$ and $B$.

Step2: Determine truth - value

The first conditional is true. But the second is false because $P$, $A$, and $B$ being collinear doesn't necessarily mean $P$ is between $A$ and $B$ (e.g., $A$ could be between $P$ and $B$). So the biconditional is false.

Step1: Write as two conditionals

Conditional 1: If segments are congruent, then their lengths are equal. Conditional 2: If the lengths of segments are equal, then the segments are congruent.

Step2: Determine truth - value

Both conditionals are true based on the definition of congruent segments. So the biconditional is true.

Step1: Write as two conditionals

Conditional 1: If two planes never intersect, then they are parallel. Conditional 2: If two planes are parallel, then they never intersect.

Step2: Determine truth - value

Both conditionals are true based on the properties of planes in geometry. So the biconditional is true.

Answer:

Conditionals: "If point $P$ is between $A$ and $B$, then $P$, $A$, and $B$ are collinear" and "If $P$, $A$, and $B$ are collinear, then point $P$ is between $A$ and $B$"; False

25.