Answer
1. By the definition of an angle bisector, because AC bisects BAD, angles 1 and 2 are congruent.
2. Because sides opposite congruent angles are congruent, BC and CD are congruent.
3. AC is congruent to itself by identity.
4. Thus, while the picture is deceiving, ABC is congruent to ACD by SAS.
5. Because CPCTC, AB is congruent to AD.
6. Because CPCTC, BC is congruent to CD.
7. Angle ACB is obtuse, so AB is longer than BC.
8. Thus, by the transitive property, AD is longer than CD.
Work Step by Step
.Given that ray AC is an angle bisector of $\angle BAD $
•$\angle 1 =\angle2 $ by the definition of angle bisector.
• taking the external angle of triangle ABC so $\angle ACD= \angle 1 + \angle ABC $ ( the external angle of a triangle is equal the sum of the two interior non adjacent angles ) .
• by theorem $ \angle ACD \gt \angle 1 $ and $\angle ACD \gt \angle ABC $
• by substituting $\angle 1 by \angle 2 $ since they are congruent so that $ \angle ACD \gt \angle 2 $.
• the segment opposite to $ \angle ACD is \overline{AD} $ and the segment opposite to $ \angle 2 is \overline{CD} $ therefore by substitution $\overline{AD} \gt CD. $
$\square $
