a Can you draw, for example, two different isosceles triangles with the same perimeter?
B
b What does congruent actually mean?
A
a False.
Type
Statement
True or false
Conditional
If two triangles have the same perimeter, then they are congruent.
False
Converse
If two triangles are congruent, then they have the same perimeter.
True
Inverse
If two triangles do not have the same perimeter, then they are not congruent.
True
Contrapositive
If two triangles are not congruent, they do not have the same perimeter.
False
B
b True.
Practice makes perfect
a Congruent triangles are identical. Therefore, if two triangles are congruent, they will have the same perimeter. However, we can create two triangles with the same perimeter but different shapes. Consider the following two isosceles triangles.
The triangles have the same perimeter, 10, but different shapes which makes them not congruent. Therefore, the conditional statement is false.
Conditional statement
To rewrite the statement using the converse, inverse and contrapositive, let's first state the hypothesis and conclusion of the conditional statement:
Hypothesis: &If two triangles have
&the same perimeter,
Conclusion: &then they are congruent
Converse
To write the converse of a conditional statement, exchange the hypothesis and the conclusion:
Hypothesis: &If two triangles are congruent,
Conclusion: &then they have the
&same perimeter
This statement is true.
Inverse
To write the inverse of a conditional statement, negate both the hypothesis and the conclusion:
Hypothesis: &If two triangles do not have
&the same perimeter,
Conclusion: &then they are not congruent
This statement is true.
Contrapositive
To write the contrapositive of a conditional statement, first write the converse. Then negate both the hypothesis and the conclusion.
Hypothesis: &If two triangles are
¬ congruent
Conclusion: &then they do not have
&the same perimeter
This statement is false.
b As we already stated, when two triangles are congruent, they are identical. This applies to everything about the triangles, including their area. Therefore, the statement is true.
