If two angles and any side of a triangle are equal to the corresponding angles and side of another triangle, then prove that the two triangles are congruent.


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Answer:


Step by Step Explanation:
  1. Let ^@ \triangle ABC ^@ and ^@ \triangle DEF ^@ be the two triangles with ^@ BC = EF, \angle A = \angle D, \text{ and } \angle B = \angle E ^@.
      A B C D E F
  2. We have to prove that @^\triangle ABC \cong \triangle DEF@^
  3. We know that the sum of the angles of a triangle is ^@180^ \circ^@. @^ \begin{aligned} \implies \angle A + \angle B + \angle C = \angle D + \angle E + \angle F = 180^ \circ && \ldots (1) \end{aligned}@^ It is given that ^@ \angle A = \angle D \text{ and } \angle B = \angle E. ^@
    Thus, from equation (1), we conclude that ^@ \angle C = \angle F. ^@
  4. In ^@ \triangle ABC \text{ and } \triangle DEF^@, we have @^ \begin{aligned} &BC = EF && \text{[By step 1]}\\ &\angle C = \angle F && \text{[By step 3]} \\ &\angle B = \angle E && \text{[By step 1]} \\ &\therefore \triangle ABC \cong \triangle DEF && \text{[By ASA criterion]} \end{aligned} @^
  5. Thus, if two angles and any side of a triangle are equal to the corresponding angles and side of another triangle, the two triangles are congruent.

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