Chapter 1.8, Problem 8PSB

a.

To determine

The converse, the inverse and the contrapositive of the given statement.

Answer to Problem 8PSB

Converse: “If a ray divides an angle into two congruent angles, then it bisects the angle.”

Inverse: “If a ray does not bisects an angle, then it does not divide the angle into two congruent lines.”

Contrapositive: “If a ray does not divide an angle into two congruent angles, then it does not bisect the angle.”

Explanation of Solution

Given information:

The sentence is “If a ray bisects an angle, it divides the angle into two congruent angles.”

Every conditional statement “If p, then q” has three other statements associated with it.

A converse (If q, then p)

An inverse $(If~p,then~q.)$

A contrapositive $(If~q,then~p.)$

By referring to above associated statements, we can write the converse, the inverse and the contrapositive of a statement.

b.

To determine

The converse, inverse, and contrapositive of “If two sides of a triangle are congruent, then angles opposite to those sides are congruent”

Answer to Problem 8PSB

Converse: “If two angles of a triangle are congruent, then sides opposite to those angles are congruent.”

Inverse: “If two sides of a triangle are not congruent, then the angles opposite to those sides are not congruent.”

Contrapositive: “If two angles of a triangle are not congruent angles, then the sides opposite to those angles are not congruent.”

Explanation of Solution

Given information:

The given sentence is “If two sides of a triangle are congruent, then angles opposite to those sides are congruent.”

Every conditional statement “If p, then q” has three other statements associated with it.

A converse (If q, then p)

An inverse $(If~p,then~q.)$

A contrapositive $(If~q,then~p.)$

By referring to above associated statements, we can write the converse, the inverse and the contrapositive of a statement.