Question

2.1 conditional statements (pp. 107-120)
learning target: understand and write conditional statements.
write the if-then form, the converse, the inverse, the contrapositive, and the biconditional of the conditional statement.

1. two lines intersect in a point.
2. 4x + 9 = 21 because x = 3.
3. supplementary angles sum to 180°.
4. right angles are 90°.

decide whether the statement about the diagram is true. explain your answer using definitions you have learned.

1. s is the midpoint of \\(\overline{ef}\\).
2. \\(\overline{es} \cong \overline{st}\\)
3. \\(\overline{st}\\) is a segment bisector of \\(\overline{ef}\\).

(diagram: e---s---f on a line, t above s)

Explanation:

To solve these problems, we'll address each one by one, focusing on understanding conditional statements and diagram - related statements.

Problem 1: "Two lines intersect in a point."

If - then form:

If two lines are intersecting, then they intersect in a point.

Converse:

If two lines intersect in a point, then the two lines are intersecting.

Inverse:

If two lines are not intersecting, then they do not intersect in a point.

Contrapositive:

If two lines do not intersect in a point, then the two lines are not intersecting.

Biconditional:

Two lines intersect if and only if they intersect in a point.

Problem 2: "$4x + 9 = 21$ because $x = 3$"

First, we verify the statement. Substitute $x = 3$ into the left - hand side of the equation $4x+9$:
$4\times3 + 9=12 + 9 = 21$, which is equal to the right - hand side.

If - then form:

If $x = 3$, then $4x + 9 = 21$.

Converse:

If $4x + 9 = 21$, then $x = 3$. (We can solve $4x+9 = 21$: $4x=21 - 9=12$, so $x = 3$)

Inverse:

If $x
eq3$, then $4x + 9
eq21$.

Contrapositive:

If $4x + 9
eq21$, then $x
eq3$.

Biconditional:

$4x + 9 = 21$ if and only if $x = 3$.

Problem 3: "Supplementary angles sum to $180^{\circ}$"

If - then form:

If two angles are supplementary, then their sum is $180^{\circ}$.

Converse:

If the sum of two angles is $180^{\circ}$, then the two angles are supplementary.

Inverse:

If two angles are not supplementary, then their sum is not $180^{\circ}$.

Contrapositive:

If the sum of two angles is not $180^{\circ}$, then the two angles are not supplementary.

Biconditional:

Two angles are supplementary if and only if their sum is $180^{\circ}$.

Problem 4: "Right angles are $90^{\circ}$"

If - then form:

If an angle is a right angle, then the measure of the angle is $90^{\circ}$.

Converse:

If the measure of an angle is $90^{\circ}$, then the angle is a right angle.

Inverse:

If an angle is not a right angle, then the measure of the angle is not $90^{\circ}$.

Contrapositive:

If the measure of an angle is not $90^{\circ}$, then the angle is not a right angle.

Biconditional:

An angle is a right angle if and only if the measure of the angle is $90^{\circ}$.

Problem 5: "S is the midpoint of $\overline{EF}$"

From the diagram, we can see that the number of segments from $E$ to $S$ and from $S$ to $F$ are equal (assuming the tick marks represent equal - length segments). By the definition of a midpoint: A midpoint of a segment is a point that divides the segment into two congruent (equal - length) segments. Since $ES=SF$ (from the diagram's segment markings), $S$ is the midpoint of $\overline{EF}$. So the statement is true.

Problem 6: "$\overline{ES}\cong\overline{ST}$"

From the diagram, $\overline{ES}$ is a horizontal segment on the line $\overleftrightarrow{EF}$, and $\overline{ST}$ is a non - horizontal segment. Visually, their lengths are not the same. Also, there is no indication (such as tick marks) that they are congruent. So the statement $\overline{ES}\cong\overline{ST}$ is false.

Problem 7: "$\overline{ST}$ is a segment bisector of $\overline{EF}$"

A segment bisector of a segment is a line, ray, or segment that intersects the segment at its midpoint. We know from problem 5 that $S$ is the midpoint of $\overline{EF}$, and $\overline{ST}$ passes through $S$. So by the definition of a segment bisector, $\overline{ST}$ is a segment bisector of $\overline{EF}$. The statement is true.

Answer:

To solve these problems, we'll address each one by one, focusing on understanding conditional statements and diagram - related statements.

Problem 1: "Two lines intersect in a point."

If - then form:

If two lines are intersecting, then they intersect in a point.

Converse:

If two lines intersect in a point, then the two lines are intersecting.

Inverse:

If two lines are not intersecting, then they do not intersect in a point.

Contrapositive:

If two lines do not intersect in a point, then the two lines are not intersecting.

Biconditional:

Two lines intersect if and only if they intersect in a point.

Problem 2: "$4x + 9 = 21$ because $x = 3$"

First, we verify the statement. Substitute $x = 3$ into the left - hand side of the equation $4x+9$:
$4\times3 + 9=12 + 9 = 21$, which is equal to the right - hand side.

If - then form:

If $x = 3$, then $4x + 9 = 21$.

Converse:

If $4x + 9 = 21$, then $x = 3$. (We can solve $4x+9 = 21$: $4x=21 - 9=12$, so $x = 3$)

Inverse:

If $x
eq3$, then $4x + 9
eq21$.

Contrapositive:

If $4x + 9
eq21$, then $x
eq3$.

Biconditional:

$4x + 9 = 21$ if and only if $x = 3$.

Problem 3: "Supplementary angles sum to $180^{\circ}$"

If - then form:

If two angles are supplementary, then their sum is $180^{\circ}$.

Converse:

If the sum of two angles is $180^{\circ}$, then the two angles are supplementary.

Inverse:

If two angles are not supplementary, then their sum is not $180^{\circ}$.

Contrapositive:

If the sum of two angles is not $180^{\circ}$, then the two angles are not supplementary.

Biconditional:

Two angles are supplementary if and only if their sum is $180^{\circ}$.

Problem 4: "Right angles are $90^{\circ}$"

If - then form:

If an angle is a right angle, then the measure of the angle is $90^{\circ}$.

Converse:

If the measure of an angle is $90^{\circ}$, then the angle is a right angle.

Inverse:

If an angle is not a right angle, then the measure of the angle is not $90^{\circ}$.

Contrapositive:

If the measure of an angle is not $90^{\circ}$, then the angle is not a right angle.

Biconditional:

An angle is a right angle if and only if the measure of the angle is $90^{\circ}$.

Problem 5: "S is the midpoint of $\overline{EF}$"

From the diagram, we can see that the number of segments from $E$ to $S$ and from $S$ to $F$ are equal (assuming the tick marks represent equal - length segments). By the definition of a midpoint: A midpoint of a segment is a point that divides the segment into two congruent (equal - length) segments. Since $ES=SF$ (from the diagram's segment markings), $S$ is the midpoint of $\overline{EF}$. So the statement is true.

Problem 6: "$\overline{ES}\cong\overline{ST}$"

From the diagram, $\overline{ES}$ is a horizontal segment on the line $\overleftrightarrow{EF}$, and $\overline{ST}$ is a non - horizontal segment. Visually, their lengths are not the same. Also, there is no indication (such as tick marks) that they are congruent. So the statement $\overline{ES}\cong\overline{ST}$ is false.

Problem 7: "$\overline{ST}$ is a segment bisector of $\overline{EF}$"

A segment bisector of a segment is a line, ray, or segment that intersects the segment at its midpoint. We know from problem 5 that $S$ is the midpoint of $\overline{EF}$, and $\overline{ST}$ passes through $S$. So by the definition of a segment bisector, $\overline{ST}$ is a segment bisector of $\overline{EF}$. The statement is true.