Question

which of the following is an open statement?
∠abc is obtuse.
an angle with measure 40° is acute.
an angle with measure 105° is obtuse.
an angle with measure 82° is obtuse.
question 8 (5 points)
identify the converse of this statement:
if a shape has 3 sides, then its a triangle.
if a shape isnt a triangle, then it doesnt have three sides.
if a shape is a triangle, then it doesnt have three sides.
if a shape is a triangle, then it has 3 sides.
if a shape doesnt have 3 sides, then its a triangle.

Explanation:

First Question (Open Statement)

An open statement is a statement whose truth value depends on the value of a variable (or in this case, the specific angle ∠ABC, which is not defined with a measure). Let's analyze each option:

• "An angle with measure \(40^\circ\) is acute": \(40^\circ\) is less than \(90^\circ\), so this is a true statement (not open).
• "An angle with measure \(105^\circ\) is obtuse": \(105^\circ\) is greater than \(90^\circ\) and less than \(180^\circ\), so this is a true statement (not open).
• "An angle with measure \(82^\circ\) is obtuse": \(82^\circ\) is less than \(90^\circ\), so this is a false statement (not open).
• "\(\angle ABC\) is obtuse": We don't know the measure of \(\angle ABC\), so its truth value is unknown (open statement).

The converse of a conditional statement "If \(p\), then \(q\)" is "If \(q\), then \(p\)". The original statement is "If a shape has 3 sides, then it's a triangle" (\(p\): shape has 3 sides; \(q\): it's a triangle). So the converse should be "If a shape is a triangle, then it has 3 sides". Let's check the options:

• "If a shape isn't a triangle, then it doesn't have three sides": This is the contrapositive (not converse).
• "If a shape is a triangle, then it doesn't have three sides": This is false and not the converse.
• "If a shape is a triangle, then it has 3 sides": This matches the converse definition.
• "If a shape doesn't have 3 sides, then it's a triangle": This is incorrect logic.

Answer:

\(\boldsymbol{\angle ABC}\) is obtuse.

Second Question (Converse of the Statement)