Question

problem 23
given: \\( \triangle b c d \cong \triangle e f d \\),
b is the midpoint
of \\( \overline{a c} \\).
prove: abef is a
parallelogram.

1. click here to insert
2. corresponding parts

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

1. b is the midpoint of

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

1. click here to insert
2. click here to insert
3. click here to insert

Explanation:

Step1: Use CPCTC

Since \(\triangle BCD\cong\triangle EFD\), by Corresponding Parts of Congruent Triangles are Congruent (CPCTC), \(BC = EF\).

Step2: Use mid - point property

Given \(B\) is the midpoint of \(\overline{AC}\), by the definition of mid - point, \(AB=BC\).

Step3: Substitute

Since \(AB = BC\) and \(BC=EF\), then \(AB = EF\) (by substitution property).

Step4: Use CPCTC for angles

Also, from \(\triangle BCD\cong\triangle EFD\), \(\angle CBD=\angle EFD\) (CPCTC). So, \(AB\parallel EF\) (if alternate interior angles are equal, then lines are parallel).

Answer:

1. \(BC = EF\)
2. Definition of mid - point
3. \(AB = EF\) and \(AB\parallel EF\); If a quadrilateral has one pair of opposite sides that are both equal and parallel, then the quadrilateral is a parallelogram.