Question

6-86.
write a converse for each conditional statement below. then, assuming the original statement is true,
decide if the converse must be always, sometimes or never true.

a. if the base angles of a triangle are congruent, then it is isosceles.

b. if a figure is a triangle, then the sum of the angles in the figure is $180^\circ$.

c. if i clean my room, then my mom will be happy.

Explanation:

For each conditional statement "If P, then Q", the converse is "If Q, then P". We then evaluate the truthfulness of the converse based on definitions, logical rules, or real-world context.

• Part a: Use the definition of isosceles triangles, which states that isosceles triangles have congruent base angles.
• Part b: Use the angle sum property; only triangles have an angle sum of 180°, no other polygon does.
• Part c: Consider real-world context: a mom could be happy for other reasons besides a clean room, or not happy even if the room is clean.

Answer:

a. Converse: If a triangle is isosceles, then its base angles are congruent.
Truth value: Always true
b. Converse: If the sum of the angles in a figure is 180°, then the figure is a triangle.
Truth value: Always true
c. Converse: If my mom is happy, then I cleaned my room.
Truth value: Sometimes true