Question

b is the midpoint of \\(\overline{ac}\\). which statement best describes the relationship between triangles abd and cbd?

Explanation:

Identify given geometric properties

Using the Geometric Proofs knowledge point
We are given:

• \(AD = 15\) and \(CD = 15\), so \(AD \cong CD\).
• \(B\) is the midpoint of \(\overline{AC}\), which means \(AB = CB\), so \(AB \cong CB\).

Apply the reflexive property

Using the Geometric Proofs knowledge point

• The segment \(\overline{BD}\) is shared by both triangles \(\triangle ABD\) and \(\triangle CBD\).
• By the Reflexive Property of Congruence, \(BD \cong BD\).

Determine triangle congruence

Using the Geometric Proofs knowledge point

• We have established three pairs of congruent sides:

1. \(AD \cong CD\) (Side)
2. \(AB \cong CB\) (Side)
3. \(BD \cong BD\) (Side)

• Therefore, \(\triangle ABD \cong \triangle CBD\) by the Side-Side-Side (SSS) Congruence Postulate.

Answer:

The triangles \(\triangle ABD\) and \(\triangle CBD\) are congruent by the Side-Side-Side (SSS) Congruence Postulate because \(AD = CD = 15\), \(AB = CB\) (since \(B\) is the midpoint of \(\overline{AC}\)), and they share the side \(\overline{BD}\) (\(BD = BD\)).