Chapter 4: Q5. (page 146)

Given: $\bar{M B} ≅ \bar{N C}; \bar{B N} ⊥ \bar{A C}; \bar{C M} ⊥ \bar{A B}$
Prove: $\bar{C M} ≅ \bar{B N}$

Short Answer

Expert verified

Two column proof is

Statement	Reason
1. $m ∠ C N B = m ∠ B M C$	Each
2. $\bar{C B} ≅ \bar{B C}$	Reflexive property
3. $\bar{M B} ≅ \bar{N C}$	Given
4. $Δ C N B ≅ Δ B M C$	HL Theorem
5. $\bar{B N} ≅ \bar{C M}$	Corresponding Parts of Congruent Triangle

Step by step solution

- Consider triangles ΔCNB and ΔBMC

As $m ∠ B M C = 90 °$ and $m ∠ C N B = 90 °$ , so both triangles are right angle triangle.

$\bar{B C}$ is hypotenuse in both triangles $Δ B M C and Δ C N B$

- Use reflexive property

In triangles $Δ C N B and Δ B M C$ , by using reflexive property common side of two triangle is congruent. So $\bar{C B} ≅ \bar{B C}$

- Prove ΔCNB≅ΔBMC

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

Using $H L$ 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, $Δ C N B ≅ Δ B M C$

- Prove BN¯≅CM¯

As corresponding parts of congruent triangles are congruent so $\bar{B N} ≅ \bar{C M}$

Thus, two column proof is

Statement	Reason
1. $m ∠ C N B = m ∠ B M C$	Each
2. $\bar{C B} ≅ \bar{B C}$	Reflexive property
3. $\bar{M B} ≅ \bar{N C}$	Given
4. $Δ C N B ≅ Δ B M C$	HL Theorem
5. $\bar{B N} ≅ \bar{C M}$	Corresponding Parts of Congruent Triangle