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Example Question #1 : How To Find If Acute / Obtuse Triangles Are Congruent
You are given triangles and
, with
and
. Which of these statements, along with what you are given, is not enough to prove that
?
I) and
have the same perimeter
II)
III)
None of these statements is enough to prove congruence.
Statement I only
Statement II only
Any of the three statements is enough to prove congruence.
Statement III only
Statement III only
If and
have the same perimeter,
, and
, it follows that
. The three triangles have the same sidelengths, setting the conditions for the Side-Side-Side Congruence Postulate.
If , then, since the sum of the degree measures of both triangles is the same (180 degrees), it follows that
. Since
and
are congruent included angles of congruent sides, this sets the conditions for the SAS Congruence Postulate.
In both of the above cases, it follows that .
However, similarly to the previous situation, if , then it follows that
, meaning that we have congruent sides and congruent nonincluded angles. However, this is not sufficient to prove congruence.
"Statement III" is the correct response.
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