What is the converse of the theorem statement: The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long.
Prove: The segment joining the midpoints of two sides of a triangle is parallel to the third side.
The converse of the theorem statement "The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long" is:
Given a triangle, if a segment connecting the midpoints of two sides of the triangle is parallel to the third side and is half as long, then the triangle is divided into two smaller triangles of equal area by this segment.
To find the converse of a theorem statement, you need to switch the hypothesis and the conclusion.
The given theorem statement is: "The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long."
The hypothesis is: "The segment connecting the midpoints of two sides of a triangle."
The conclusion is: "is parallel to the third side and is half as long."
To form the converse, we switch the hypothesis and conclusion, resulting in the following statement:
"If a segment connecting two sides of a triangle is parallel to the third side and is half as long, then it connects the midpoints of the two sides."
That is the converse of the given theorem statement.