Question

given: \\( \overline{\mathrm{jk}} \cong \overline{\mathrm{lk}} ; \angle \mathrm{mjk} \cong \angle \mathrm{mlk} ; \overline{\mathrm{km}} \\) bisects \\( \overline{\mathrm{jl}} \\).
prove: \\( \overline{\mathrm{km}} \\) bisects \\( \angle \mathrm{jkl} \\).
complete the statements in the two - column proof.
enter your answers in the boxes.

statements	reasons
2. \\( \overline{\mathrm{km}} \\) bisects \\( \overline{\mathrm{jl}} \\).	2. given
3. \\( \overline{\mathrm{mj}} \cong \overline{\mathrm{ml}} \\)	3. definition of segment bisector
4. \\( \triangle \mathrm{jkm} \cong \triangle \mathrm{lkm} \\)	4. sas
5. \\( \angle \square \cong \angle \square \\)	5. cpctc
6. \\( \overline{\mathrm{km}} \\) bisects \\( \angle \mathrm{jkl} \\).	6. definition of angle bisector

Explanation:

Step1: Recall CPCTC

CPCTC (Corresponding Parts of Congruent Triangles are Congruent) means that if two triangles are congruent, then their corresponding parts (angles and sides) are congruent.

Step2: Identify corresponding angles

Since \(\triangle JKM\cong\triangle LKM\) (from statement 4), and we want to show that \(KM\) bisects \(\angle JKL\) (which is composed of \(\angle 2\) and \(\angle 4\) in the diagram). The angles \(\angle 2\) and \(\angle 4\) are the angles at vertex \(K\) for triangles \(\triangle JKM\) and \(\triangle LKM\) respectively.

Answer:

\(\angle 2\cong\angle 4\)