Question

question 8 of 10

look at the diagram below.

if \\(\angle abd \cong \angle cbd\\), then \\(ad = cd\\).

a. true
b. false

Explanation:

Identify given geometric properties

We are given a diagram with ray \(BD\) between rays \(BA\) and \(BC\).
The segments \(DA\) and \(DC\) are perpendicular to rays \(BA\) and \(BC\) respectively:

$$ \angle DAB = 90^\circ, \quad \angle DCB = 90^\circ $$

We are also given that \(\angle ABD \cong \angle CBD\).

Apply triangle congruence criteria

Using the Locus of Points concept, we analyze the two right triangles \(\triangle DAB\) and \(\triangle DCB\):

• \(\angle DAB \cong \angle DCB\) (both are \(90^\circ\))
• \(\angle ABD \cong \angle CBD\) (given)
• Segment \(BD\) is shared (reflexive property: \(BD \cong BD\))

By the Angle-Angle-Side (AAS) congruence theorem:

$$ \triangle DAB \cong \triangle DCB $$

Conclude corresponding parts

Since the triangles are congruent, their corresponding parts are congruent (CPCTC):

$$ AD = CD $$

Thus, the statement "If \(\angle ABD \cong \angle CBD\), then \(AD = CD\)" is true. This is also a direct application of the Angle Bisector Theorem.

Answer:

• A. True (Correct answer)
• B. False