Question

statements

1. \\( \triangle a b c \\) with ( \overleftrightarrow { d e } \parallel \overline { a c } \\)
2. ( \angle d b a \cong \angle b, \angle e b c \cong \angle c \\)
3. ( m \angle d b a = m \angle a ; m \angle e b c = m \angle c \\)
4. ( m \angle d b a + m \angle a b c + m \angle e b c = 180 ^ { \circ } \\)
5. ( m \angle a + m \angle a b c + m \angle c = 180 ^ { \circ } \\)

(1 point)
statement 2 has a mistake. it should say ( \angle d b a \cong \angle b, \angle e b c \cong \angle a \\)
statement 4 has a mistake. it should say ( m \angle d b a + m \angle a b c + m \angle a c b = 180 ^ { \circ } \\)
statement 2 has a mistake. it should say ( \angle d b a \cong \angle a, \angle e b c \cong \angle c \\)
statement 4 has a mistake. it should say ( m \angle a + m \angle a b c + m \angle c = 180 ^ { \circ } \\)

Explanation:

Step1: Analyze Statement 2

When \( \overleftrightarrow{DE}\parallel\overline{AC}\), by the alternate - interior angles theorem, \( \angle DBA\cong\angle A\) (not \( \angle B\)) because \( \angle DBA\) and \( \angle A\) are alternate - interior angles formed by the transversal \( AB\) intersecting the parallel lines \( \overleftrightarrow{DE}\) and \( \overline{AC}\). And \( \angle EBC\cong\angle C\) (correct as given, since \( \angle EBC\) and \( \angle C\) are alternate - interior angles formed by the transversal \( BC\) intersecting the parallel lines \( \overleftrightarrow{DE}\) and \( \overline{AC}\)).

Step2: Analyze Statement 4

A straight - angle has a measure of \( 180^{\circ}\). The angles \( \angle DBA\), \( \angle ABC\), and \( \angle EBC\) form a straight - angle. So \( m\angle DBA + m\angle ABC+m\angle EBC = 180^{\circ}\) is correct based on the definition of a straight - angle.

Answer:

Statement 2 has a mistake. It should say \( \angle DBA\cong\angle A,\angle EBC\cong\angle C\).