Question

Write proofs in two-column form.

Given: $\overline{\mathbf{GH}}\perp \overline{\mathbf{HJ}}$; $\overline{\mathbf{KJ}}\perp \overline{\mathbf{HJ}}$; $\overline{\mathbf{GJ}}\cong \overline{\mathbf{KH}}$.

Prove: $\overline{\mathbf{GH}}\cong \overline{\mathbf{KJ}}$

Short answer

The proof in two-column form is:

Step-by-step solution

Step 1.  Observe the given diagram.

The given diagram is:

Step 2.  Description of step

It is being given that $\overline{GH}\perp \overline{HJ}$,$\overline{KJ}\perp \overline{HJ}$and $\overline{GJ}\cong \overline{KH}$.

As, $\overline{GH}\perp \overline{HJ}$, therefore by using the definition of perpendicular lines it can be said that $m\angle GHJ=90°$.

As, $\overline{KJ}\perp \overline{HJ}$,therefore by using the definition of perpendicular lines it can be said that $m\angle KJH=90°$.

Therefore, $m\angle GHJ=m\angle KJH=90°$.

Therefore, $\angle GHJ\cong \angle KJH$.

As, $\overline{GJ}\cong \overline{KH}$, therefore,it can be noticedthat the hypotenuses of the triangles $▵GHJ$ and $▵KJH$ are equal.

In the triangles$▵GHJ$ and $▵KJH$, it can be noticed that the side $HJ$ is common.

Therefore, $\overline{HJ}\cong \overline{HJ}$by using the reflexive property.

As, $\overline{HJ}\cong \overline{HJ}$, therefore, it can be noticed thatthe legs of the triangles$▵GHJ$ and $▵KJH$ are equal.

In the triangles $▵GHJ$ and $▵KJH$, it can be noticed that $\overline{GJ}\cong \overline{KH}$ and $\overline{HJ}\cong \overline{HJ}$.

Therefore, the triangles $▵GHJ$ and $▵KJH$ are congruent triangles by HL postulate.

The triangles $▵GHJ$ and $▵KJH$ are congruent triangles.

Therefore, by using the corresponding parts of congruent triangles it can be said that $\overline{GH}\cong \overline{KJ}$.

Step 3.  Write the proof in two-column form.

The proof in two-column form is: