Question

given: \\( \triangle abc \\).
d is the midpoint
of \\( \overline { a b } \\) and
e is the midpoint
of \\( \overline { a c } \\).
prove: \\( \overline { d e } \parallel \overline { b c } \\) and
\\( d e = \frac { 1 } { 2 } b c \\).

1. bcfd is a

parallelogram.

1. \\( \overline { c } \parallel \overline { b c } \\) and \\( d f = b c \\)

Explanation:

Step1: Definition of parallelogram

If a quadrilateral is a parallelogram, then opposite sides are parallel and equal. Since \(BCFD\) is a parallelogram (statement 13), by the definition of a parallelogram, \(DF\parallel BC\) and \(DF = BC\).

Step2: Mid - segment property

We know that \(D\) is the mid - point of \(AB\) and \(E\) is the mid - point of \(AC\). Also, from the parallelogram \(BCFD\), we can show the relationship between \(DE\) and \(BC\). But for statement 13, the reason is "Definition of parallelogram". For statement 14, since \(BCFD\) is a parallelogram, the reason is "Opposite sides of a parallelogram are parallel and equal"

Answer:

1. Reason: Definition of parallelogram
2. Reason: Opposite sides of a parallelogram are parallel and equal