Question

consider this diagram of triangle \\( \delta a b c \\).
it is given that \\( \overline{c d} \\) bisects \\( \angle a c b \\) and that \\( \delta a b c \\) is isosceles with \\( \overline{a c} \cong \overline{b c} \\).
complete the proof that \\( \angle c a d \cong \angle c b d \\).
\\( \overline{c d} \\) bisects \\( \angle a c b \\) given
\\( \delta a b c \\) is isosceles with \\( \overline{a c} \cong \overline{b c} \\). given
\\( \overline{c d} \cong \overline{c d} \\) select
\\( \angle a c d \cong \angle b c d \\) select
\\( \delta c a d \cong \delta c b d \\) select
\\( \angle c a d \cong \angle c b d \\) select

Explanation:

Step1: Reason for $\overline{CD}\cong\overline{CD}$

Any segment is congruent to itself. This is the reflexive property of congruence.

Step2: Reason for $\angle ACD\cong\angle BCD$

Since $\overline{CD}$ bisects $\angle ACB$, by the definition of an angle bisector, it divides the angle into two congruent angles.

Step3: Reason for $\triangle CAD\cong\triangle CBD$

We have $\overline{AC}\cong\overline{BC}$ (given), $\angle ACD\cong\angle BCD$ (from angle - bisector), and $\overline{CD}\cong\overline{CD}$ (reflexive property). So, by the Side - Angle - Side (SAS) congruence criterion, $\triangle CAD\cong\triangle CBD$.

Step4: Reason for $\angle CAD\cong\angle CBD$

If two triangles are congruent, then their corresponding parts (angles in this case) are congruent. This is the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem.

Answer:

• For $\overline{CD}\cong\overline{CD}$: Reflexive Property of Congruence
• For $\angle ACD\cong\angle BCD$: Definition of Angle Bisector
• For $\triangle CAD\cong\triangle CBD$: Side - Angle - Side (SAS) Congruence Criterion
• For $\angle CAD\cong\angle CBD$: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)