Question

Given:$\overline{\mathbf{M}\mathbf{B}}\mathbf{\cong }\overline{\mathbf{N}\mathbf{C}}\mathbf{;}\overline{\mathbf{B}\mathbf{N}}\mathbf{\perp }\overline{\mathbf{A}\mathbf{C}}\mathbf{;}\overline{\mathbf{C}\mathbf{M}}\mathbf{\perp }\overline{\mathbf{A}\mathbf{B}}$

Prove:$\overline{\mathbf{C}\mathbf{M}}\mathbf{\cong }\overline{\mathbf{B}\mathbf{N}}$

Short answer

Two column proof is

Statement	Reason
1.$m\angle CNB=m\angle BMC$	Each
2.$\overline{CB}\cong \overline{BC}$	Reflexive property
3.$\overline{MB}\cong \overline{NC}$	Given
4.$\Delta CNB\cong \Delta BMC$	HL Theorem
5.$\overline{BN}\cong \overline{CM}$	Corresponding Parts of Congruent Triangle

Step-by-step solution

- Consider triangles ΔCNB and ΔBMC

As$m\angle BMC=90°$ and $m\angle CNB=90°$, so both triangles are right angle triangle.

$\overline{BC}$is hypotenuse in both triangles$\Delta BMC\text{and}\Delta CNB$

- Use reflexive property

In triangles $\Delta CNB\text{and}\Delta BMC$, by using reflexive property common side of two triangle is congruent. So$\overline{CB}\cong \overline{BC}$

- Prove ΔCNB≅ΔBMC

As hypotenuse and one leg of both right angle triangle is congruent to each other.

Using $HL$Theorem “If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent.”

Thus,$\Delta CNB\cong \Delta BMC$

- Prove BN¯≅CM¯

As corresponding parts of congruent triangles are congruent so$\overline{BN}\cong \overline{CM}$

Thus, two column proof is

Statement	Reason
1.$m\angle CNB=m\angle BMC$	Each
2.$\overline{CB}\cong \overline{BC}$	Reflexive property
3.$\overline{MB}\cong \overline{NC}$	Given
4.$\Delta CNB\cong \Delta BMC$	HL Theorem
5.$\overline{BN}\cong \overline{CM}$	Corresponding Parts of Congruent Triangle