Question 66216: Triangle ABC and triangle PQR each have two sides whose lengths are 7 inches and an angle whose measure is 40 degrees. Are the triangles congruent?
Found 3 solutions by venugopalramana, ikleyn, math_tutor2020:
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! Triangle ABC and triangle PQR each have two sides whose lengths are 7 inches and an angle whose measure is 40 degrees. Are the triangles congruent?
YES THEOREM IS SAS
IF 40 DEG.IS THE INCLUDED ANGLE,THEN SAS APPLIES SINCE
AB=PQ=7
AC=PR=7
ANGLE BAC=ANGLE QPR=40
IF 40 DEG. IS BASE ANGLE THEN , THE OTHER BASE ANGLE=40(SINCE THE TRIANGLE IS ISOCLLES) AND HENCE THE REMAINIG ANGLE = INCLUDED ANGLE =180-40-40=100 IN BOTH TRIANGLES.HENCE AGAIN SAS APPLIES.
Answer by ikleyn(54062)  (Show Source):
You can put this solution on YOUR website! .
Triangle ABC and triangle PQR each have two sides whose lengths are 7 inches and an angle whose measure is 40 degrees.
Are the triangles congruent?
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The answer "Yes" in the post by @venugopalramana is WRONG.
The correct answer is "Not necessary" or "it depends".
There are three cases:
(1) The triangles ABC and PQR are congruent if the 40°-angles are the angles between their sides of 7 inches,
according to the SAS test of congruence for triangles.
(2) The triangles ABC and PQR are congruent if the mentioned 40°-angles are their base angles,
according to the same SAS test of congruence for triangles.
Indeed, in this case the triangles have corresponding congruent base angles, which implies that
their angles between the sides of 7 inches are also congruent
(because they have then the same measure of 180° - 40° - 40° = 100°).
(3) The triangles ABC and PQR are  congruent, if one triangle has the 40°-angle between two 7 inches legs,
while another triangle has the 40°-angle as a base angle.
Then, obviously, the triangles have different angles between the 7 inches sides, so they are not congruent.
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I'm surprised to see that tutor @venugopalramana got confused by such a simple question.
Answer by math_tutor2020(3852)  (Show Source):
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