Question

5
select the correct answer.
given: line i and line m intersect
prove: \\( \angle 1 \cong \angle 3 \\)
which statement and reason best completes the proof?
\\( \bigcirc \\) a. 3. \\( \mathrm { m } \angle 1 + \mathrm { m } \angle 2 = 180 ^ { \circ } \\) 3. definition of supplementary angles
\\( \bigcirc \\) b. 3. \\( \mathrm { m } \angle 1 + \mathrm { m } \angle 3 = 180 ^ { \circ } \\) 3. definition of supplementary angles
\\( \bigcirc \\) c. 3. \\( \angle 3 \\) is supplementary to \\( \angle 2 \\) 3. linear pair theorem
\\( \bigcirc \\) d. 3. \\( \angle 1 \\) is supplementary to \\( \angle 4 \\) 3. linear pair theorem

Explanation:

Step1: Analyze each option

• Option A: If we only know \(m\angle1 + m\angle2=180^{\circ}\), we can't directly use the congruent supplements theorem to prove \(\angle1\cong\angle3\). Because the congruent supplements theorem requires two angles that are supplements of the same angle.
• Option B: \(m\angle1 + m\angle3 = 180^{\circ}\) is not based on the given figure and the linear - pair relationship. In the given figure, \(\angle1\) and \(\angle2\), \(\angle2\) and \(\angle3\) have linear - pair relationships.
• Option C:
• By the linear - pair theorem, \(\angle3\) and \(\angle2\) form a linear pair (since line \(l\) and line \(m\) intersect). So, \(\angle3\) is supplementary to \(\angle2\) (statement).
• The reason is the linear - pair theorem (reason).
• Now, since \(\angle1\) is supplementary to \(\angle2\) (from step 2) and \(\angle3\) is supplementary to \(\angle2\), by the congruent supplements theorem (\(If\angle A+\angle C = 180^{\circ}\) and \(\angle B+\angle C=180^{\circ}\), then \(\angle A\cong\angle B\)), we can prove \(\angle1\cong\angle3\).
• Option D: \(\angle1\) is supplementary to \(\angle4\) is a correct linear - pair relationship, but it doesn't help in proving \(\angle1\cong\angle3\) using the congruent supplements theorem with the information about \(\angle2\).

Answer:

C. 3. \(\angle3\) is supplementary to \(\angle2\); 3. linear pair theorem