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

Given : \overline{J K} ⊥ \overline{J M}, \overline{K L} ⊥ \overline{M L}, Given : ∠ J ≅ ∠ M, ∠ K ≅ ∠ L Prove : \overline{J M} ⊥ \overline{M L} and \overline{J K} ⊥ \overline{K L}

Let's now consider the given diagram.

Recall the definition of perpendicular lines. Since we know that $\overline{J K}$ is perpendicular to $\overline{J M}$ and that $\overline{K L}$ is perpendicular to $\overline{M L},$ we can conclude that both $∠ J$ and $∠ L$ are right angles. This means that they measure $9 0^{\circ} .$

By the definition of perpendicular lines $m ∠ J = 9 0^{\circ}$ and $m ∠ L = 9 0^{\circ} .$

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

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

We can now find the measures of

∠ M

and

∠ L

by using the Transitive Property of Equality.

{m ∠ J = m ∠ M m ∠ J = 9 0^{\circ} \Rightarrow m ∠ M = 9 0^{\circ} and {m ∠ K = m ∠ L m ∠ L = 9 0^{\circ} \Rightarrow m ∠ K = 9 0^{\circ}

We found that both

∠ M

and

∠ K

measure

9 0^{\circ} .

By the Transitive Property of Equality $m ∠ M = 9 0^{\circ}$ and $m ∠ K = 9 0^{\circ} .$

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, $\overline{J M}$ is perpendicular to $\overline{M L},$ and $\overline{J K}$ is pependicular to $\overline{K L} .$

By the definition of perpendicularity, $\overline{J M} ⊥ \overline{M L}$ and $\overline{J K} ⊥ \overline{K L} .$

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.

Given : \overline{J K} ⊥ \overline{J M}, \overline{K L} ⊥ \overline{M L}, Given : ∠ J ≅ ∠ M, ∠ K ≅ ∠ L Prove : \overline{J M} ⊥ \overline{M L} and \overline{J K} ⊥ \overline{K L}

Proof. By the definition of perpendicular lines, the measure of

∠ J

and

∠ L

is

9 0^{\circ} .

Next, by the definition of congruent angles,

∠ J

and

∠ M

have the same measure, and

∠ K

and

∠ L

have the same measure. By the Transitive Property of Equality,

∠ M

and

∠ K

measure

9 0^{\circ} .

Then, by the definition of perpendicular lines,

\overline{J M}

is perpendicular to

\overline{M L}

and

\overline{J K}

is perpendicular to

\overline{K L} .

Exercise

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