Question

In a scalene triangle \( A B C \) is similar to triangle \( D E F \). Which statement is \( f a l s e \) ? 1) \( A B: B C=D E: E F \) 2) \( A C: D F=B C: E F \) 3) \( \angle A C B \cong \angle D F E \) 4) \( \angle A B C \cong \angle E D F \)

Ask by Pierce Rose. in China
Jan 23,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

所有选项都是正确的,没有假的陈述。

Solution

在不等边三角形 \( A B C \) 与三角形 \( D E F \) 相似的情况下,以下哪个陈述是 \( f a l s e \)? 1) \( A B: B C=D E: E F \) 2) \( A C: D F=B C: E F \) 3) \( \angle A C B \cong \angle D F E \) 4) \( \angle A B C \cong \angle E D F \) 我们来分析每个选项: 1) 由于三角形相似,边的比率是相等的,因此 \( A B: B C=D E: E F \) 是正确的。 2) 这个比率 \( A C: D F=B C: E F \) 也应该是正确的,因为相似三角形的对应边比是相等的。 3) 角 \( \angle A C B \) 和 \( \angle D F E \) 是对应角,因此它们是相等的,这个陈述也是正确的。 4) 角 \( \angle A B C \) 和 \( \angle E D F \) 是对应角,因此它们也是相等的,这个陈述也是正确的。 综上所述,所有选项都是正确的,因此没有一个选项是假的。请确认问题的描述或选项是否有误。

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Extra Insights

In similar triangles, corresponding sides are in proportion and corresponding angles are congruent. However, the false statement among the options given is statement 2) \( A C: D F=B C: E F \). This statement mistakenly claims a proportional relationship between non-corresponding sides. For similar triangles, it's essential to only compare corresponding sides, which indeed are \( A B \) to \( D E \), \( B C \) to \( E F \), and \( A C \) to \( D F \). To remember, two triangles are similar if all their corresponding angles are equal and their corresponding sides are in proportion. So, when checking similar triangles, always align the correct angles and sides to avoid tripping over false statements. If you remember that, you'll avoid common pitfalls and mathematical misinterpretations in the future!

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