Question

if \\( \angle a b d \cong \angle c b d \\) and \\( a d = 43 \\), what is \\( c d \\)?

Explanation:

Step1: Identify the triangles

Consider \(\triangle ABD\) and \(\triangle CBD\). \(\angle ABD\cong\angle CBD\) (given), \(\angle BAD = \angle BCD=90^{\circ}\) (right angles), and \(BD = BD\) (common side).

Step2: Apply AAS congruence

By the Angle - Angle - Side (AAS) congruence criterion (\(\angle ABD\cong\angle CBD\), \(\angle BAD\cong\angle BCD\), \(BD = BD\)), \(\triangle ABD\cong\triangle CBD\).

Step3: Use congruent triangles property

Since \(\triangle ABD\cong\triangle CBD\), their corresponding sides are equal. So \(AD = CD\).

Answer:

\(43\)