Question

how should the following statement be written in if - then form?
the sum of the measures of the angles of a triangle is 180 degrees.
if the sum of the measures of the angles is 180 degrees, then there exists a triangle.
if a triangle exists, then the sum of the measures of the angles is 180 degrees.
if two rays are opposite, then they form a straight line.

Explanation:

Step1: Analyze the "if - then" structure

The "if - then" form is “If \(p\), then \(q\)” where \(p\) is the hypothesis and \(q\) is the conclusion.

Step2: Identify hypothesis and conclusion for each option

• For “If two rays are opposite, then they form a straight line”: Hypothesis (\(p\)) is “two rays are opposite”, conclusion (\(q\)) is “they form a straight line”. This is about rays, not triangle angles.
• For “If a triangle exists, then the sum of the measures of the angles is 180 degrees”: Hypothesis (\(p\)) is “a triangle exists”, conclusion (\(q\)) is “the sum of the measures of the angles is 180 degrees”. But the original statement is about the sum of angles leading to triangle existence.
• For “If the sum of the measures of the angles is 180 degrees, then there exists a triangle”: Hypothesis (\(p\)) is “the sum of the measures of the angles is 180 degrees”, conclusion (\(q\)) is “there exists a triangle”. This matches the logical “if - then” structure for the given statement about the sum of triangle angles.

Answer:

If the sum of the measures of the angles is 180 degrees, then there exists a triangle.