a The converse of a statement exchanges the conclusion and hypotenuse.
B
b Consider the Triangle Angle Sum Theorem.
C
c Can your mom be happy about anything other than you cleaning your room?
A
aConverse: If a triangle is isosceles, then the base angles are congruent.
Converse true?: Yes
B
bConverse: If the sum of the angles in a figure is 180^(∘), then the figure is a triangle.
Converse true?: Yes
C
cConverse: If my mom is happy, then I have cleaned my room.
Converse true?: No
Practice makes perfect
a Let's highlight the hypothesis and conclusion in the conditional statement.
If the base angles of a triangle are congruent,
then it is isosceles.
The converse of a conditional statement, q→ p, exchanges the hypothesis and conclusion of the conditional statement.
If a triangle is isosceles,
then the base angles are congruent.
This is true by the Base Angles theorem.
b Like in Part A, we will highlight the hypothesis and conclusion in the conditional statement.
If a figure is a triangle,
then the sum of the angles in the figure is $180^(∘)$.
Again, to get the converse of the conditional statement, q→ p, we exchange the hypothesis and conclusion.
If the sum of the angles in a figure is $180^(∘)$,
then the figure is a triangle.
To show that the converse is true, consider instead what it would mean to have a sum of 180^(∘). Two angles is not a figure. A quadrilateral's angles sum to 360^(∘). Two angles is too few, and four is too many. Thus, there are three angles in the figure which means it is a triangle. Note that we are assuming that the figure is a polygon.
c Like in Parts A and B, we will highlight the hypothesis and conclusion in the conditional statement.
If I clean my room,
then my mom will be happy.
Again, to get the converse of the conditional statement, q→ p, we exchange the hypothesis and conclusion.
If my mom is happy,
then I have cleaned my room.
Your mom's happiness is not contingent on whether you cleaned your room or not. Therefore, we conclude that this is not true.
