Consider the following two statements about two triangles:
P : Two Triangles have equal area, and two sides of one are equal to two sides of the other.
Q : They are congruent.
Which of the following statements is true?
(A) P is necessary and sufficient for Q.
(B) P is necessary but not sufficient for Q.
(C) P is sufficient but not necessary for Q.
(D) P is neither necessary nor sufficient for Q.
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T a k e a g e n e r a l c a s e w h e r e t h e s i d e s o f a t r i a n g l e a r e a , b , c a n d a ′ , b , c . L e t t h e a n g l e f a c i n g a b e A a n d t h e a n g l e f a c i n g a ′ b e A ′ . A r e a = 2 1 b c s i n A = 2 1 b c s i n A ′ ⇒ s i n A = s i n A ′ . ⇒ A = A ′ O R , A = π − A ′ . S o , P n e e d n o t n e c e s s a r i l y m e a n A = A ′ . ∴ P n e e d n o t m e a n Q . B u t t h e o t h e r w a y a r o u n d , w e c a n s e e t h a t Q d o e s i n d e e d m e a n P i s s a t i s f i e d .
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If Q is true then P must be true, so P is necessary for Q .
However, P can be true without Q being true. For example, suppose we have two isosceles triangles, one with the two equal sides separated by an angle of 6 0 ∘ and one with the same two equal sides being separated by an angle of 1 2 0 ∘ . Then the two triangles have the same area, thus satisfying statement P , but they are not congruent, hence do not satisfy statement Q . Thus in this case P does not imply Q , and thus P is not sufficient for Q .
So P is necessary but not sufficient for Q , making option B the correct choice.