Let's take a look at the given information and the desired outcome of the proof.
&Given: JK⊥JM, KL⊥ML,
& ∠ J ≅ ∠ M, ∠ K≅ ∠ L
&Prove: JM⊥MLandJK⊥KL
Let's now consider the given diagram.
Recall the definition of perpendicular lines. Since we know that JK is perpendicular to JM and that KL is perpendicular to ML, we can conclude that both ∠ J and ∠ L are right angles. This means that they measure 90^(∘).
By the definition of perpendicular lines m∠ J=90^(∘) and m∠ L=90^(∘).
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=90^(∘) ⇒ m∠ M=90^(∘)
and
m∠ K= m∠ L m∠ L=90^(∘) ⇒ m∠ K=90^(∘)
We found that both ∠ M and ∠ K measure 90^(∘).
By the Transitive Property of Equality m∠ M=90^(∘) and m∠ K=90^(∘).
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, JM is perpendicular to ML, and JK is pependicular to KL.
By the definition of perpendicularity, JM⊥ ML and JK⊥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.
&Given: JK⊥JM, KL⊥ML,
& ∠ J ≅ ∠ M, ∠ K≅ ∠ L
&Prove: JM⊥MLandJK⊥KL
Proof. By the definition of perpendicular lines, the measure of ∠ J and ∠ L is 90^(∘). 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 90^(∘). Then, by the definition of perpendicular lines, JM is perpendicular to ML and JK is perpendicular to KL.
