Question

question 1-2
a true statement is given.
\if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent.\
select the statement that represents the inverse of this statement.
○ if two triangles are not congruent, then two sides and the included angle of one triangle are not congruent to two sides and the included angle of the other triangle.
○ two triangles are congruent if and only if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle.
○ if two triangles are congruent, then two sides and the included angle of one of the triangles are congruent to two sides and the included angle of the second triangle.
○ if two sides and the included angle of one triangle are not congruent to two sides and the included angle of a second triangle, then the triangles are not congruent.

Explanation:

Step1: Recall the definition of inverse statement

The inverse of a conditional statement "If \( p \), then \( q \)" is "If not \( p \), then not \( q \)". The original statement here is: "If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent." So \( p \): two sides and included angle of one triangle congruent to two sides and included angle of another triangle; \( q \): triangles are congruent.

Step2: Analyze each option

• Option 1: "If two triangles are not congruent, then two sides and the included angle of one triangle are not congruent to two sides and the included angle of the other triangle." This is the contrapositive (not inverse), since it's "If not \( q \), then not \( p \)".
• Option 2: "Two triangles are congruent if and only if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle." This is a biconditional, not an inverse.
• Option 3: "If two triangles are congruent, then two sides and the included angle of one of the triangles are congruent to two sides and the included angle of the second triangle." This is the converse (If \( q \), then \( p \)), not inverse.
• Option 4: "If two sides and the included angle of one triangle are not congruent to two sides and the included angle of a second triangle, then the triangles are not congruent." This matches the inverse definition: "If not \( p \), then not \( q \)".

Answer:

The fourth option (the one stating: "If two sides and the included angle of one triangle are not congruent to two sides and the included angle of a second triangle, then the triangles are not congruent.")