Question

problem 24
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. \\( \angle 3 \cong \angle 4, d a = c f \\)
2. corresponding parts of

congruent triangles are
congruent (c.p.c.t.c.)

1. d is the midpoint of

\\( \overline { a b } \\).

1. given
2. definition of a

midpoint

1. \\( b d = c f \\)

11.

Explanation:

Step1: Definition of midpoint

Since \(D\) is the midpoint of \(\overline{AB}\), by the definition of a midpoint, \(AD = BD\).

Step2: Substitution

We know from statement \(8\) that \(DA=CF\). Substituting \(AD\) with \(BD\) (from step 1) in the equation \(DA = CF\), we get \(BD=CF\).

Answer:

1. \(AD = BD\)
2. Substitution Property (using \(AD = BD\) from step 10 and \(DA=CF\) from step 8)