Question

if cd bisects ∠acd, what additional information could be used to prove △abc ≅ △dbc using sas? select three options.
□m∠abc = 125° and ab ≅ db
□△acd is isosceles with base ad
□△abd is isosceles with base ad
□cd = 52 cm
□ab = 29 cm

Explanation:

Step1: Recall SAS (Side - Angle - Side) criterion

SAS requires two pairs of corresponding sides equal and the included angles equal for two triangles to be congruent. In \(\triangle ABC\) and \(\triangle DBC\), we already know that \(BC = BC\) (common side). Since \(CB\) bisects \(\angle ACD\), \(\angle ACB=\angle DCB\).

Step2: Analyze option 1

If \(m\angle ABC = 125^{\circ}\) and \(AB = DB\), we don't have the correct included - angle situation for SAS. The given angle is not the included angle between the sides we are considering.

Step3: Analyze option 2

If \(\triangle ACD\) is isosceles with base \(AD\), then \(AC = CD\). Along with \(\angle ACB=\angle DCB\) (because \(CB\) bisects \(\angle ACD\)) and \(BC = BC\), we can use SAS to prove \(\triangle ABC\cong\triangle DBC\).

Step4: Analyze option 3

If \(\triangle ABD\) is isosceles with base \(AD\), it gives information about \(\triangle ABD\) but not directly relevant to proving \(\triangle ABC\cong\triangle DBC\) using SAS.

Step5: Analyze option 4

If \(CD = 52\mathrm{cm}\), and since \(AC = 52\mathrm{cm}\) (from the figure), along with \(\angle ACB=\angle DCB\) (because \(CB\) bisects \(\angle ACD\)) and \(BC = BC\), we can use SAS to prove \(\triangle ABC\cong\triangle DBC\).

Step6: Analyze option 5

If \(AB = 29\mathrm{cm}\), it doesn't help in the SAS proof as we need the side - angle - side relationship. We don't have the correct angle between the sides \(AB\) (or \(DB\)) and \(BC\) for SAS.

Answer:

\(\triangle ACD\) is isosceles with base \(AD\), \(CD = 52\mathrm{cm}\)