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.
...
the conditional statement is true.
d. if the converse statement is true, write a true biconditional statement. choose the correct answer below:
○ a. ( angle b ) is an obtuse angle if and only if ( 0^circ > mangle b > 360^circ ).
○ b. ( angle b ) is an obtuse angle if and only if ( 90^circ < mangle b < 180^circ ).
○ c. the converse statement is false.

Explanation:

To determine the correct biconditional statement, we first recall the definitions:

• The original conditional statement is "If ∠B is an obtuse angle, then \(90^\circ < m\angle B < 180^\circ\)".
• The converse of this conditional is "If \(90^\circ < m\angle B < 180^\circ\), then ∠B is an obtuse angle". Since the conditional is true (as given) and we assume the converse is true (for part d), a biconditional combines both, meaning ∠B is an obtuse angle if and only if \(90^\circ < m\angle B < 180^\circ\).

Now let's analyze the options:

• Option A has incorrect angle measures (\(0^\circ > m\angle B > 360^\circ\) is not related to obtuse angles).
• Option B correctly states the biconditional using the definition of an obtuse angle.
• Option C is incorrect because we are to assume the converse is true for part d (and the converse here is true as per the definition of an obtuse angle).

Answer:

B. \(\angle B\) is an obtuse angle if and only if \(90^{\circ}< m\angle B < 180^{\circ}\)