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?
\\( \bigcirc \\) a. \\( \angle j l m \\) is supplementary to \\( \angle k l m \\) by the transitive property.
\\( \bigcirc \\) b. \\( m \angle j l m + m \angle k l m = 180 ^ { \circ } \\) by the definition of supplementary angles.
\\( \bigcirc \\) c. \\( \angle j l m \cong \angle k l m \\) by the linear pair theorem.
\\( \bigcirc \\) d. \\( m \angle j l m = m \angle k l m \\) by the definition of congruence.

Explanation:

Step1: Recall the definition of supplementary angles

If two angles are supplementary, then the sum of their measures is \(180^{\circ}\). Since \(\angle JLM\) and \(\angle KLM\) are a linear pair (by the linear - pair theorem), we should state \(m\angle JLM + m\angle KLM=180^{\circ}\) based on the definition of supplementary angles.

Step2: Analyze each option

• Option A: The linear - pair theorem (not the transitive property) states that \(\angle JLM\) and \(\angle KLM\) are supplementary. So, this option is incorrect.
• Option B: By the definition of supplementary angles (for a linear pair \(\angle JLM\) and \(\angle KLM\)), \(m\angle JLM + m\angle KLM = 180^{\circ}\). This is the missing step.
• Option C: \(\angle JLM\cong\angle KLM\) is given, not by the linear - pair theorem. The linear - pair theorem is about the supplementary relationship of a linear pair of angles. So, this option is incorrect.
• Option D: \(m\angle JLM=m\angle KLM\) by the definition of congruence is already used in the proof after stating the angles are congruent. It is not the missing step.

Answer:

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