Question

select the correct answer.
given: \\( \angle j l m \cong \angle k l m \\)
prove: \\( m \angle j l m=90^{circ} \\)
by the linear pair theorem, \\( \angle j l m \\) is supplementary to \\( \angle k l m \\). since \\( \angle j l m \cong \angle k l m \\), by the definition of congruence, \\( m \angle j l m=m \angle k l m \\). applying the substitution property of equality, \\( m \angle j l m+m \angle j l m=180^{circ} \\). simplifying the equation, \\( m \angle j l m=90^{circ} \\).
what step is missing from this proof?
a. \\( m \angle j l m+m \angle k l m=180^{circ} \\) by the definition of supplementary angles.
b. \\( \angle j l m \\) is supplementary to \\( \angle k l m \\) by the transitive property.
c. \\( \angle j l m \cong \angle k l m \\) by the linear pair theorem.
d. \\( m \angle j l m=m \angle k l m \\) by the definition of congruence.

Explanation:

• Option A:
• By the linear pair theorem, \(\angle JLM\) and \(\angle KLM\) form a linear pair.
• The definition of supplementary angles states that if two angles form a linear pair, their measures add up to \(180^{\circ}\). So, \(m\angle JLM + m\angle KLM=180^{\circ}\) is a necessary step before substitution (since \(m\angle JLM = m\angle KLM\) from congruence) to get \(m\angle JLM+m\angle JLM = 180^{\circ}\).
• Option B:
• \(\angle JLM\) is supplementary to \(\angle KLM\) by the linear - pair theorem (not the transitive property). The transitive property is for equality or congruence (\(a = b\) and \(b = c\) implies \(a = c\)), not for supplementary angles.
• Option C:
• \(\angle JLM\cong\angle KLM\) is given, not by the linear - pair theorem. The linear - pair theorem is about supplementary angles (sum to \(180^{\circ}\)), not congruence.
• Option D:
• \(m\angle JLM = m\angle KLM\) by the definition of congruence is already used in the proof (\(m\angle JLM + m\angle JLM=180^{\circ}\) comes from substituting \(m\angle KLM\) with \(m\angle JLM\)).

Answer:

A. \(m\angle JLM + m\angle KLM = 180^{\circ}\) by the definition of supplementary angles.