Question

question 3 of 10. based only on the given information, it is guaranteed that $overline{ac}congoverline{bc}$. given: $\triangle abc$, $overline{ab}perpoverline{cd}$, $angle cadcongangle cbd$. a. true b. false

Explanation:

Step1: Identify right - angled triangles

In \(\triangle ACD\) and \(\triangle BCD\), \(\angle ADC=\angle BDC = 90^{\circ}\) since \(AB\perp CD\).

Step2: Identify equal angles

We are given that \(\angle CAD=\angle CBD\).

Step3: Identify common side

\(CD\) is common to both \(\triangle ACD\) and \(\triangle BCD\).

Step4: Apply AAS congruence

By the Angle - Angle - Side (AAS) congruence criterion, \(\triangle ACD\cong\triangle BCD\) (because \(\angle CAD=\angle CBD\), \(\angle ADC=\angle BDC\), and \(CD = CD\)).

Step5: Use congruent - triangle property

If \(\triangle ACD\cong\triangle BCD\), then corresponding sides are equal. So, \(\overline{AC}\cong\overline{BC}\).

Answer:

A. True