Question 1185643: Hello! Could you please help my child with these questions? Would appreciate if could please include an explanation for each answer.
Please determine if the statements below are ALWAYS , SOMETIMES or NEVER True. Please justify your answer.
If AB≅BC in ΔABC, then ∠BAC≅∠ABC.Always, sometimes, never?
If ∠BAC≅∠ABC in ΔABC, then AB≅BC. Always, sometimes, never?
CPCTC can be used as a reason in a proof for proving triangles congruent. Always, sometimes, never?
Answer by ikleyn(52784)   (Show Source):
You can put this solution on YOUR website! .
Hello! Could you please help my child with these questions? Would appreciate if could please include an explanation for each answer.
Please determine if the statements below are ALWAYS , SOMETIMES or NEVER True. Please justify your answer.
If AB≅BC in ΔABC, then ∠BAC≅∠ABC.Always, sometimes, never?
If ∠BAC≅∠ABC in ΔABC, then AB≅BC. Always, sometimes, never?
CPCTC can be used as a reason in a proof for proving triangles congruent. Always, sometimes, never?
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(1) If AB≅BC in ΔABC, then ∠BAC≅∠ABC. Always, sometimes, never?
If AB≅BC in ΔABC, it means that AB and BC are lateral sides of the isosceles triangle ABC
with the vertex B opposite to the base side.
In this situation, the Geometry theorem states that the base angles A and C are congruent.
But the question asks about congruency of the OTHER pair of angles, namely A and B.
These two angles, A and B, are not necessary congruent.
Under given condition, they are congruent ONLY for a particular case of the EQUILATERAL triangle.
THEREFORE, the ONLY answer to this question is "SOMETIMES".
(2) If ∠BAC≅∠ABC in ΔABC, then AB≅BC. Always, sometimes, never?
If ∠BAC≅∠ABC in ΔABC, it means that A and B are the base angles in the isosceles triangle ABC,
whose lateral sides are AC and AB. Then the Geometry theorem states that the lateral sides AC and BC are congruent.
But the question asks about congruency of different pair of sides, namely AB and BC.
These sides are not necessary congruent.
Under the given conditions, they are congruent ONLY for a particular case of the EQUILATERAL triangle.
THEREFORE, the ONLY answer to this question is "SOMETIMES".
(3) CPCTC can be used as a reason in a proof for proving triangles congruent. Always, sometimes, never?
CPCTC is this logical statement "If two triangles are congruent, then their corresponding sides are congruent".
The posed question asks if CPCTC can be used in the proof of congruency triangles, i.e. opposite statement.
The answer is "NEVER".
Solved.
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Please let me know if the son does understand my solution.
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