Exercise 24 Page 485

Let's take a look at the given information and the desired outcome of the proof. Let's now consider the given diagram.

Recall the definition of perpendicular lines. Since we know that $\overline{JK}$ is perpendicular to $\overline{JM}$ and that $\overline{KL}$ is perpendicular to $\overline{ML},$ we can conclude that both $\angle J$ and $\angle L$ are right angles. This means that they measure $90^{\circ }.$

By the definition of perpendicular lines $m\angle J=90^{\circ }$ and $m\angle L=90^{\circ }.$

Next, we are given that $\angle J$ and $\angle M$ are congruent angles, and that $\angle K$ and $\angle L$ are also congruent angles. Therefore, we can state that the measures of $\angle J$ and $\angle M$ are equal, as well as the measures of $\angle K$ and $\angle L.$

By the definition of congruent angles $m\angle J=m\angle M$ and $m\angle K=m\angle L.$

We can now find the measures of $\angle M$ and $\angle L$ by using the Transitive Property of Equality. We found that both $\angle M$ and $\angle K$ measure $90^{\circ }.$

By the Transitive Property of Equality $m\angle M=90^{\circ }$ and $m\angle K=90^{\circ }.$

Finally, we will use the definition of perpendicularity one last time. If an angle between two segments or lines is a right angle, then these segments or lines are said to be perpendicular. Therefore, $\overline{JM}$ is perpendicular to $\overline{ML},$ and $\overline{JK}$ is pependicular to $\overline{KL}.$

By the definition of perpendicularity, $\overline{JM}\perp \overline{ML}$ and $\overline{JK}\perp \overline{KL}.$

Now let's gather all of the statements we wrote to formulate a paragraph proof.

Completed Proof

Considering the given information, we can summarize all the steps in a paragraph proof. Proof. By the definition of perpendicular lines, the measure of $\angle J$ and $\angle L$ is $90^{\circ }.$ Next, by the definition of congruent angles, $\angle J$ and $\angle M$ have the same measure, and $\angle K$ and $\angle L$ have the same measure. By the Transitive Property of Equality, $\angle M$ and $\angle K$ measure $90^{\circ }.$ Then, by the definition of perpendicular lines, $\overline{JM}$ is perpendicular to $\overline{ML}$ and $\overline{JK}$ is perpendicular to $\overline{KL}.$