Question

1. if the given statement is not in if - then form, rewrite it. write inverse, converse and contrapositive of the given conditional statement. determine the truth value of all four statements. if a statement is false, give a counterexample.

a. if two angles are vertical, then they are congruent.
converse:
inverse:
contrapositive:
b. soccer players are athletes.
conditional:
converse:
inverse:
contrapositive:

Explanation:

A.

Converse:

If two angles are congruent, then they are vertical.
This is false. Counter - example: Two angles of \(30^{\circ}\) in two different triangles are congruent but not vertical.

Inverse:

If two angles are not vertical, then they are not congruent.
This is false. Counter - example: Two non - vertical angles of \(90^{\circ}\) (right angles) are congruent.

Contrapositive:

If two angles are not congruent, then they are not vertical.
This is true.

B.

Conditional:

If a person is a soccer player, then the person is an athlete.
This is true.

Converse:

If a person is an athlete, then the person is a soccer player.
This is false. Counter - example: A basketball player is an athlete but not a soccer player.

Inverse:

If a person is not a soccer player, then the person is not an athlete.
This is false. Counter - example: A tennis player is not a soccer player but is an athlete.

Contrapositive:

If a person is not an athlete, then the person is not a soccer player.
This is true.

Answer:

A.

Converse:

If two angles are congruent, then they are vertical.
This is false. Counter - example: Two angles of \(30^{\circ}\) in two different triangles are congruent but not vertical.

Inverse:

If two angles are not vertical, then they are not congruent.
This is false. Counter - example: Two non - vertical angles of \(90^{\circ}\) (right angles) are congruent.

Contrapositive:

If two angles are not congruent, then they are not vertical.
This is true.

B.

Conditional:

If a person is a soccer player, then the person is an athlete.
This is true.

Converse:

If a person is an athlete, then the person is a soccer player.
This is false. Counter - example: A basketball player is an athlete but not a soccer player.

Inverse:

If a person is not a soccer player, then the person is not an athlete.
This is false. Counter - example: A tennis player is not a soccer player but is an athlete.

Contrapositive:

If a person is not an athlete, then the person is not a soccer player.
This is true.