Question

2.1 conditional statements (pp. 107–120)
watch
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 $180^circ$.
4. right angles are $90^circ$.

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

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

diagram: horizontal line with points $e$, $s$, $f$ ( $e$---$s$---$f$ ) and $t$ above $s$

Explanation:

For conditional statements, convert to if-then form, then derive converse (switch hypothesis/conclusion), inverse (negate both), contrapositive (negate and switch), and biconditional (if and only if). For diagram questions, use definitions: midpoint (divides segment into two congruent parts), congruent segments (equal length), segment bisector (passes through midpoint).

Answer:

1. If-then: If two lines intersect, then they intersect in a point. Converse: If two lines intersect in a point, then they intersect. Inverse: If two lines do not intersect, then they do not intersect in a point. Contrapositive: If two lines do not intersect in a point, then they do not intersect. Biconditional: Two lines intersect if and only if they intersect in a point.
2. If-then: If x = 3, then 4x + 9 = 21. Converse: If 4x + 9 = 21, then x = 3. Inverse: If x ≠ 3, then 4x + 9 ≠ 21. Contrapositive: If 4x + 9 ≠ 21, then x ≠ 3. Biconditional: 4x + 9 = 21 if and only if x = 3.
3. If-then: If two angles are supplementary, then they sum to 180°. Converse: If two angles sum to 180°, then they are supplementary. Inverse: If two angles are not supplementary, then they do not sum to 180°. Contrapositive: If two angles do not sum to 180°, then they are not supplementary. Biconditional: Two angles are supplementary if and only if they sum to 180°.
4. If-then: If an angle is a right angle, then its measure is 90°. Converse: If an angle measures 90°, then it is a right angle. Inverse: If an angle is not a right angle, then its measure is not 90°. Contrapositive: If an angle does not measure 90°, then it is not a right angle. Biconditional: An angle is a right angle if and only if its measure is 90°.
5. True. By diagram, S divides EF into two congruent segments, so S is the midpoint (def. of midpoint).
6. False. ES (on EF) and ST (vertical from S) have no indicated congruent length (no congruency marks).
7. True. Since S is the midpoint of EF, ST passes through the midpoint, so it is a segment bisector (def. of segment bisector).