Question

consider the statement \an obtuse angle, b, measures ( 90^circ < mangle b < 180^circ ).\
a. write the statement as a conditional statement.
b. if the conditional statement is true, write the converse of the conditional.
c. if the conditional statement is false, give a counterexample for the conditional.
d. if the converse statement is true, write a true biconditional statement.
e. if the converse statement is false, give a counterexample for the converse.
...
b. if the conditional statement is true, which of the following is the converse of the conditional?
a. if ( 90^circ < mangle b < 180^circ ), then ( angle b ) is not an obtuse angle.
b. if ( 90^circ < mangle b < 180^circ ), then ( angle b ) is an obtuse angle.
c. if ( 90^circ > mangle b > 180^circ ), then ( angle b ) is an obtuse angle.
d. the conditional statement is false.

Explanation:

Part a:

Step1: Identify hypothesis and conclusion

The original statement is "An obtuse angle, \( B \), measures \( 90^\circ < m\angle B < 180^\circ \)". The hypothesis (condition) is "an angle \( B \) is obtuse" and the conclusion (result) is "\( 90^\circ < m\angle B < 180^\circ \)".

Step2: Form conditional statement

A conditional statement is in the form "If [hypothesis], then [conclusion]". So, "If \( \angle B \) is an obtuse angle, then \( 90^\circ < m\angle B < 180^\circ \)".

Part b:

Step1: Recall converse definition

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

Step2: Apply to our conditional

From part a, the conditional is "If \( \angle B \) is an obtuse angle, then \( 90^\circ < m\angle B < 180^\circ \)". So the converse is "If \( 90^\circ < m\angle B < 180^\circ \), then \( \angle B \) is an obtuse angle". Among the options, option B matches this.

Part c:

The conditional statement "If \( \angle B \) is an obtuse angle, then \( 90^\circ < m\angle B < 180^\circ \)" is true (by definition of obtuse angle). So we don't need a counterexample here.

Part d:

Answer:

B. If \( 90^{\circ}