Let's consider each of the statements one at a time using the given

p

and

q .

p = q = Two angles are supplementary The measures of the angles sum to 18 0^{\circ}

p \to q,

in an if-then form.

If t w o a n g l e s a r e s u p p l e m e n t a r y, then t h e m e a s u r e s o f t h e a n g l e s s u m t o 18 0^{\circ} .

By the definition of supplementary angles we know that they have a sum of

18 0^{\circ},

so this is true.

The converse of a conditional statement,

q \to p,

exchanges the hypothesis and the conclusion of the conditional statement.

If t h e m e a s u r e s o f t w o a n g l e s s u m t o 18 0^{\circ}, then t h e t w o a n g l e s a r e s u p p l e m e n t a r y .

By the definition of supplementary angles, we know the converse is also true.

The inverse of a conditional statement,

\sim p \to \sim q,

requires us to negate the hypothesis and the conclusion of the conditional statement.

If t w o a n g l e s a r e not s u p p l e m e n t a r y, then t h e m e a s u r e s o f t h e a n g l e s do not sum to 18 0^{\circ} .

Again, the definition of supplementary angles tells us that the angles have a sum of

18 0^{\circ} .

Therefore, if two angles do not sum to

18 0^{\circ},

they cannot be supplementary. This is a true statement.

The contrapositive of a conditional statement,

\sim q \to \sim p,

starts out with the converse of the conditional statement. Then we have to negate the hypothesis and the conclusion.

If t h e m e a s u r e s o f t w o a n g l e s do not s u m t o 18 0^{\circ}, then t h e t w o a n g l e s a r e not s u p p l e m e n t a r y .

By the same logic used to know that the inverse is true, we know that the contrapositive is true.

Exercise