Question

Write a proof for exercise 8 in two-column form.

Short answer

The two-column proof is:

Statement	Reason
∠1is supplementary to∠3	Given
∠2is supplementary to∠3	Given
∠2 + ∠3 = 180	Definition of supplementary angles
∠1 + ∠3 = 180	Definition of supplementary angles
∠1 = ∠2	Subtraction property of equality
∠1≅∠2	Definition of congruent angles.
	Theorem of the converse of perpendicular lines

Step-by-step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Definition of supplementary angles.

The supplementary angles are the angles whose sum is 180°.

Step 3. Write the proof of the statement j⊥k.

∠1 is supplementary to∠3.

Therefore, the sum of angles ∠1 and ∠3 is 180°.

That implies, $\angle 1+\angle 3=180°\left(1\right)$

∠2 is supplementary to∠3.

Therefore, the sum of angles ∠2 and ∠3 is 180°.

That implies,

$\angle 2+\angle 3=180°\left(2\right)$

From the equations (1) and (2), it is obtained that:

$\begin{array}{c}\angle 1+\angle 3=\angle 2+\angle 3\\ \angle 1=\angle 2\end{array}$

That implies $\angle 1\cong \angle 2$.

From the given diagram it can be noticed that the angles∠1 and∠2 are the adjacent angles.

Therefore, the angles∠1 and∠2 are the adjacent congruent angles.

Now by using the theorem of the converse of perpendicular lines it can be said that if the angles ∠1 and ∠2 are the adjacent congruent angles, then the lines j and k are perpendiculars.

That implies $j\perp k$.

Step 4. Write the two-column proof.

Statement	Reason
∠1is supplementary to∠3	Given
∠2is supplementary to∠3	Given
∠2 + ∠3 = 180	Definition of supplementary angles
∠1 + ∠3 = 180	Definition of supplementary angles
∠1 = ∠2	Subtraction property of equality
∠1≅∠2	Definition of congruent angles.
	The theorem of the converse of perpendicular lines