Question

applying theorems about isosceles triangles
cd bisects ∠acb. which statements must be true? choose three correct
answers.
m∠cda = m∠cdb
ac = cd
ad = bd
m∠dca = m∠dac
m∠acd = m∠bcd

Explanation:

Step1: Analyze the angle - bisector property

Since \(CD\) bisects \(\angle ACB\), by the definition of an angle bisector, \(m\angle ACD=m\angle BCD\).

Step2: Use the property of isosceles triangles (if \(AC = BC\) as indicated by the tick - marks)

In \(\triangle ACD\) and \(\triangle BCD\), \(AC = BC\) (given by the tick - marks), \(m\angle ACD=m\angle BCD\) (from the angle - bisector), and \(CD=CD\) (common side). So, \(\triangle ACD\cong\triangle BCD\) (by the Side - Angle - Side (SAS) congruence criterion).

Step3: Use the congruence result

If \(\triangle ACD\cong\triangle BCD\), then \(AD = BD\) (corresponding parts of congruent triangles are equal) and \(m\angle CDA=m\angle CDB\) (corresponding parts of congruent triangles are equal).

Answer:

\(m\angle CDA = m\angle CDB\), \(AD = BD\), \(m\angle ACD = m\angle BCD\)