Question

n
\\( \angle a o m \cong \angle d o n \\)
given
\\( \angle a o m \cong \angle b o m \\)
transitive
property of
congruence
\\( \angle c o n \cong \angle b o m \\)
transitive
property of
congruence
\\( \angle b o m \cong \angle d o n \\)
vertical angles
theorem
\\( \angle a o m \cong \angle c o n \\)
vertical angles
theorem
the student says that the proof shows that \\( \angle a o m \cong \angle b o m \\) because they are vertical angles.
which statement best corrects the students interpretation?
o a. \\( \angle a o m \cong \angle b o m \\) because they are complementary, not because they are vertical angles.
o b. \\( \angle c o n \cong \angle b o m \\) because they are vertical angles.
o c. \\( \angle a o m \cong \angle b o m \\) because they are supplementary, not because they are vertical angles.
o d. \\( \angle a o m \cong \angle b o m \\) because they are both congruent to the same angle, not because they are vertical angles.

Explanation:

Step1: Analyze the given proof

The proof uses the transitive property of congruence. We know that \(\angle AOM\cong\angle DON\) (given) and \(\angle BOM\cong\angle DON\) (vertical angles theorem). By the transitive property of congruence (\(a\cong b\) and \(b\cong c\) implies \(a\cong c\)), \(\angle AOM\cong\angle BOM\) because they are both congruent to \(\angle DON\), not because they are vertical angles.

Step2: Check each option

• Option A: There is no information in the proof about \(\angle AOM\) and \(\angle BOM\) being complementary.
• Option B: The question is about correcting the interpretation of \(\angle AOM\cong\angle BOM\), not about \(\angle CON\cong\angle BOM\)
• Option C: The student's original statement is wrong as \(\angle AOM\) and \(\angle BOM\) are not vertical angles.
• Option D: Since \(\angle AOM\cong\angle DON\) and \(\angle BOM\cong\angle DON\), by transitive property \(\angle AOM\cong\angle BOM\) (they are both congruent to the same angle \(\angle DON\))

Answer:

D. \(\angle AOM\cong\angle BOM\) because they are both congruent to the same angle, not because they are vertical angles.