Question

practice and problem-solving exercises
practice
each conditional statement below is true. write its converse. if the converse is
also true, combine the statements as a biconditional.

1. if two segments have the same length, then they are congruent.
2. algebra if ( x = 12 ), then ( 2 x - 5 = 19 ).
3. if a number is divisible by 20, then it is even.
4. algebra if ( x = 3 ), then ( | x | = 3 ).
5. in the united states, if it is july 4, then it is independence day.
6. if ( p

ightarrow q ) is true, then ( sim q
ightarrow sim p ) is true.
write the two statements that form each biconditional.

1. a line bisects a segment if and only if the line intersects the segment only at

its midpoint.

1. an integer is divisible by 100 if and only if its last two digits are zeros.
2. you live in washington, d.c., if and only if you live in the capital of the

united states.

1. a polygon is a triangle if and only if it has exactly three sides.
2. an angle is a right angle if and only if it measures 90.

Explanation:

Step1: Understand the concept of converse

The converse of a conditional statement "If \(p\), then \(q\)" is "If \(q\), then \(p\)".

Step2: Analyze each conditional statement

• For statement 7:
• Original: If two segments have the same length, then they are congruent (\(p\): two segments have the same length; \(q\): they are congruent).
• Converse: If two segments are congruent, then they have the same length. This converse is true.
• Biconditional: Two segments are congruent if and only if they have the same length.
• For statement 8:
• Original: If \(x = 12\), then \(2x-5 = 19\) (\(p\): \(x = 12\); \(q\): \(2x - 5=19\)).
• Converse: If \(2x-5 = 19\), then \(x = 12\). Solving \(2x-5=19\) gives \(2x=24\), so \(x = 12\). The converse is true.
• Biconditional: \(x = 12\) if and only if \(2x-5 = 19\).
• For statement 9:
• Original: If a number is divisible by 20, then it is even (\(p\): a number is divisible by 20; \(q\): it is even).
• Converse: If a number is even, then it is divisible by 20. For example, 2 is even but not divisible by 20. The converse is false.
• For statement 10:
• Original: If \(x = 3\), then \(|x|=3\) (\(p\): \(x = 3\); \(q\): \(|x| = 3\)).
• Converse: If \(|x|=3\), then \(x = 3\). But \(|x| = 3\) implies \(x=\pm3\). The converse is false.
• For statement 11:
• Original: In the United States, if it is July 4, then it is Independence Day (\(p\): it is July 4; \(q\): it is Independence Day).
• Converse: In the United States, if it is Independence Day, then it is July 4. This converse is true.
• Biconditional: In the United States, it is July 4 if and only if it is Independence Day.
• For statement 12:
• Original: If \(p

ightarrow q\) is true, then \(\sim q
ightarrow\sim p\) is true (\(p\): \(p
ightarrow q\) is true; \(q\): \(\sim q
ightarrow\sim p\) is true).

• Converse: If \(\sim q

ightarrow\sim p\) is true, then \(p
ightarrow q\) is true. Since \(p
ightarrow q\) and \(\sim q
ightarrow\sim p\) are logically equivalent (contrapositive), the converse is true.

• Biconditional: \(p

ightarrow q\) is true if and only if \(\sim q
ightarrow\sim p\) is true.

Step3: Analyze biconditional statements (13 - 17)

• For statement 13:
• Two statements:
• If a line bisects a segment, then the line intersects the segment only at its midpoint.
• If a line intersects a segment only at its midpoint, then the line bisects the segment.
• For statement 14:
• Two statements:
• If an integer is divisible by 100, then its last two digits are zeros.
• If an integer's last two digits are zeros, then it is divisible by 100.
• For statement 15:
• Two statements:
• If you live in Washington, D.C., then you live in the capital of the United States.
• If you live in the capital of the United States, then you live in Washington, D.C.
• For statement 16:
• Two statements:
• If a polygon is a triangle, then it has exactly three sides.
• If a polygon has exactly three sides, then it is a triangle.
• For statement 17:
• Two statements:
• If an angle is a right angle, then it measures \(90^{\circ}\).
• If an angle measures \(90^{\circ}\), then it is a right angle.

Answer:

• Statement 7:
• Converse: If two segments are congruent, then they have the same length. Biconditional: Two segments are congruent if and only if they have the same length.
• Statement 8:
• Converse: If \(2x - 5=19\), then \(x = 12\). Biconditional: \(x = 12\) if and only if \(2x-5 = 19\).
• Statement 9:
• Converse: If a number is even, then it is divisible by 20. (Converse is false)
• Statement 10:
• Converse: If \(|x| = 3\), then \(x = 3\). (Converse is false)
• Statement 11:
• Converse: In the United States, if it is Independence Day, then it is July 4. Biconditional: In the United States, it is July 4 if and only if it is Independence Day.
• Statement 12:
• Converse: If \(\sim q

ightarrow\sim p\) is true, then \(p
ightarrow q\) is true. Biconditional: \(p
ightarrow q\) is true if and only if \(\sim q
ightarrow\sim p\) is true.

• Statement 13:
• Two statements: (1) If a line bisects a segment, then the line intersects the segment only at its midpoint. (2) If a line intersects a segment only at its midpoint, then the line bisects the segment.
• Statement 14:
• Two statements: (1) If an integer is divisible by 100, then its last two digits are zeros. (2) If an integer's last two digits are zeros, then it is divisible by 100.
• Statement 15:
• Two statements: (1) If you live in Washington, D.C., then you live in the capital of the United States. (2) If you live in the capital of the United States, then you live in Washington, D.C.
• Statement 16:
• Two statements: (1) If a polygon is a triangle, then it has exactly three sides. (2) If a polygon has exactly three sides, then it is a triangle.
• Statement 17:
• Two statements: (1) If an angle is a right angle, then it measures \(90^{\circ}\). (2) If an angle measures \(90^{\circ}\), then it is a right angle.