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.
...
e. if the converse statement is false, which of the following is a counterexample for the converse?
a. a right angle ( \theta ) with measure ( 90^circ < mangle \theta < 180^circ ).
b. an acute angle ( \theta ) with measure ( 90^circ < mangle \theta < 180^circ ).
c. the converse statement is true.

Explanation:

Part a: Write as conditional statement

Step1: Identify hypothesis and conclusion

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

Step2: Form conditional

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

Part b: Write converse if conditional is true

Step1: Recall converse definition

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

Step2: Apply to our conditional

Our conditional is "If an angle \( B \) is obtuse, then \( 90^\circ < m\angle B < 180^\circ \)". So the converse is "If \( 90^\circ < m\angle B < 180^\circ \), then angle \( B \) is obtuse".

Part c: Counterexample if conditional is false (but our conditional is true, so this step is not needed here as the conditional about obtuse angles is a standard definition, so it's true)

Part d: Biconditional if converse is true

Step1: Recall biconditional definition

A biconditional is " \( p \) if and only if \( q \)" when both conditional and converse are true.

Step2: Form biconditional

Since both the conditional (part a) and its converse (part b) are true (because the definition of an obtuse angle is an angle with measure between \( 90^\circ \) and \( 180^\circ \)), the biconditional is "An angle \( B \) is obtuse if and only if \( 90^\circ < m\angle B < 180^\circ \)".

Part e: Counterexample for converse (but converse is true, however let's analyze the options)

Answer:

s:
a. If an angle \( B \) is obtuse, then \( 90^\circ < m\angle B < 180^\circ \).
b. If \( 90^\circ < m\angle B < 180^\circ \), then angle \( B \) is obtuse.
d. An angle \( B \) is obtuse if and only if \( 90^\circ < m\angle B < 180^\circ \).
e. C. The converse statement is true.